Charles University in PragueFa ulty of S ien eBa helor's Thesis

Jan HermannTheoreti al Investigation of SilverClusters in ZeolitesDepartment of Physi al and Ma romole ular ChemistrySupervisor: do . RNDr. Petr Na htigall, Ph.D.Degree programme: ChemistryPrague 2011

Title Theoreti al Investigation of Silver Clusters in ZeolitesAuthor Jan HermannDepartment Department of Physi al and Ma romole ular ChemistrySupervisor do . RNDr. Petr Na htigall, Ph.D.Keywords silver lusters, zeolites, ele troni transitionsAbstra t The stru ture and stability of Agn+3 lusters (n = 1, 2, 3) in zeolite Yand their photo hemi al properties are investigated. Silver-ex hanged zeo-lites have potential te hnologi al appli ations as UV-to-visible light onvert-ers. The stru ture and stability of silver lusters were examined using theperiodi density fun tional theory (DFT), and employing a developed andprogrammed geometry optimization driver. Non-redu ed silver ations arepreferably lo ated in 6-member rings, far from ea h other. Upon a partialredu tion, trigonal lusters are formed in the sodalite ages. The photo hem-i al properties are examined by various methods in luding the EOM-CCSD,MRPT2 and TD-DFT. No harge transfer transitions from ligands to sil-ver lusters are found. Instead, transitions involving d and s ele trons onsilver lusters fall in the visible spe tral range. Our results ontradi t theprevious suggestion proposing existen e of linear silver lusters through adouble-6-member ring unit. Both parts of our investigations (stru tural andphoto hemi al) disprove this previous proposition. A new interpretationof experimental spe tra is proposed�the redu ed trigonal silver luster isresponsible for photoa tivity of Ag-doped zeolites with a low silver loading.Theoreti al investigation reported in this ba helor's thesis represents the �rstattempt for a systemati study of photo hemi al properties of silver lustersin zeolites.Název prá e Teoreti ké studium klastr· st°íbra v zeolite hAutor Jan HermannKatedra Katedra fyzikální a makromolekulární hemieVedou í bakalá°ské prá e do . RNDr. Petr Na htigall, Ph.D.Klí£ová slova klastry st°íbra, zeolity, elektroni ké p°e hodyAbstrakt Prá e se zabývá strukturou a stabilitou klastr· Agn+3 (n = 1, 2, 3)v zeolitu Y a jeji h foto hemi kými vlastnostmi. Zeolity s iontov¥ vym¥n¥-ným st°íbrem mají poten iální te hnologi ké vyuºití jako konvertory UV naviditelné sv¥tlo. Struktura a stabilita klastr· st°íbra je vy²et°ována metodoufunk ionálu hustoty (DFT) a periodi kým modelem s pomo í vyvinutéhoa naprogramovaného algoritmu pro optimaliza i geometrie. Neredukovanést°íbrné kationty se p°ednostn¥ na házejí v ²esti£lenný h kruzí h, daleko odsebe. Po £áste£né reduk i se tvo°í trigonální klastry uvnit° sodalitový h jed-notek. Foto hemi ké vlastnosti jsou vy²et°ovány r·znými metodami v£etn¥EOM-CCSD, MRPT2 a TD-DFT. Nejsou nalezeny ºádné elektroni ké p°e- hody z ligand· na klastry st°íbra. Naopak p°e hody d a s elektron· klastr·st°íbra spadají do viditelného spektra. Ob¥ £ásti na²í studie (strukturální ifoto hemi ká) vyvra ejí d°ív¥j²í návrh na existen i lineární h klastr· st°íb-ra skrz dvojitý ²esti£lenný kruh. Místo n¥j navrhujeme jinou interpreta iexperimentální h spekter�za fotoaktivitou st°íbrem dotovaného zeolitu Ys nízkým obsahem Ag stojí redukovaný trigonální klastr st°íbra. Teoreti kývýzkum popsaný v této bakalá°ské prá i p°edstavuje první pokus o systema-ti kou studii foto hemi ký h vlastností klastr· st°íbra v zeolite h.

I would like to thank Petr Na htigall for many helpful dis ussions aboutele troni stru ture al ulations and zeolites, and for ommenting this thesis;Ota Bludský for his help in developing the geometry optimization algorithm;Luká² Graj iar for providing the Al distribution in zeolite Y; and MiroslavRube² for helping me to over ome initial troubles with Linux and variousquantum- hemistry programs.

I de lare that I arried out this ba helor's thesis independently, and that Iused only the ited literature and other sour es.No part of this thesis has been used to obtain any a ademi degree yet.Prague, June 2, 2011 Jan Hermann

ContentsIntrodu tion 11 Geometry optimization 41.1 Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61.1.1 Quadrati approximation . . . . . . . . . . . . . . . . . 61.1.2 Line sear h . . . . . . . . . . . . . . . . . . . . . . . . 81.1.3 Internal redundant oordinates . . . . . . . . . . . . . 91.2 Algorithm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 131.3 Implementation . . . . . . . . . . . . . . . . . . . . . . . . . . 152 Silver lusters in zeolite Y 192.1 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 232.2 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 283 Ex ited states of silver lusters 303.1 Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 313.2 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 323.2.1 Transitions . . . . . . . . . . . . . . . . . . . . . . . . 343.3 Dis ussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37Con lusions 39Bibliography 41A Program listing 44

Introdu tionZeolites are mi roporous materials with ri h and yet well-de�ned stru ture.They onsist of inter onne ted tetravalent SiO4 and AlO �4 tetrahedra whi hform ompli ated topologies with various avities and pores. Despite their omplex geometry, zeolites are rystals and their stru ture an be des ribedby means of rystallography.Presen e of negatively- harged aluminium tetrahedra in the zeolite frame-work has to be ompensated by extra-framework ations to maintain ele -troneutrality. The size of the mi ropores and avities is su� ient for vari-ous mole ules to enter. This pro ess gives rise to two main appli ations ofzeolites�adsorption and atalysis.A silver ation has an ele tron on�guration of [Kr℄ 4d10. Upon its redu -tion, bulk silver forms as 5s orbitals ombine into a half-o upied ondu tionband. However, when lusters onsisting of only several silver atoms areformed, the 5s states remain dis rete. This leads to interesting photo hemi- al properties and appli ations.It is di� ult to study silver lusters experimentally be ause they areunstable with respe t to aggregation. One possibility to ir umvent thisproblem is to form them inside a zeolite. Lowered mobility of the ationsand steri onstraints imposed by the zeolite framework prevent the lustersfrom aggregation. Zeolites A, Rho, X, Y, habazite and mordenite have beenused for this purpose.An advantage of a silver ation is that it an be introdu ed into the zeoliteby simple ion-ex hange and subsequently partially redu ed. This provides ontrol over the size and harge of the silver lusters.The interest in silver-ex hanged zeolites started in 1962 when it was foundthat hydrated Ag-A hanges olor reversibly from white to yellow upon de-hydration. A partial redu tion of silver ations and formation of lusters wasproposed as a me hanism for this olor hange. Various silver lusters havebeen suggested in the literature sin e then: o tahedral Ag 06 , linear Ag 2+3 , y li Ag+3 , Ag+5 , et . However, neither was de�nitively proved and the iden-tity of silver parti les inside zeolites still remains a puzzle [1℄.1

It has been only re ently shown that upon thermal treatment or photoa -tivation, silver-ex hanged zeolites an be ome powerful �uores ent emittersin the UV�visible range [2�4℄. This behaviour has a great potential in te hno-logi al appli ations be ause UV-to-visible onverters are needed in �uores entlights.Currently, materials based on rare-earth elements are used for onvertingUV into blue, green and red light, the ombination of whi h gives the desiredwhite light. But this te hnology has several �aws. The used ompounds arevery s ar e and expensive, and produ ed mainly by China. Their miningbears extreme e ologi al burden. Final produ ts (energy saving lights) areneither re y lable, nor biodegradable. Also, the generated white light on-sists of sharp spe tral lines whi h is not the desired property as it renderseverything rather ontrast-less. Potential Ag/zeolite�based light bulb ouldbe better in all these aspe ts. Additionally, it would be heap and its pro-du tion would not require mu h initial investment as zeolites are ommonlyused in te hnology today.Silver-doped zeolites has also been suggested as possible data-storage orlabel materials. With the use of laser, individual sodalite ages of zeoliteA an be lit 3D-sele tively and the en oded information an be re overedwithout errors [5℄.To bring a zeolite-based light from theory to reality, several requirementsmust be a omplished. Namely, the UV/Vis onversion must be e� ientenough and the spe tral hara teristi s must satisfy legislative standards. Itis known from experiment that the opti al properties an be tuned by hang-ing Si/Al ratio, silver load, hoi e of se ondary ation, zeolite topology orsynthesis of the zeolite. However, a systemati optimization of these parame-ters is ompli ated by the fa t that it is not understood whi h silver parti lesexist in zeolites and what kind of ele troni transitions o ur in them. Theaim of this work is to ontribute to the hara terization of silver spe ies inzeolites and to investigate their opti al properties. To a hieve this goal, thework has been divided into three parts.1. A key step of the study is to �nd whi h silver parti les are formedinside zeolites. To properly model zeolites, quantum- hemi al programs thatemploy periodi ity of the rystal latti e have to be used. However, geometryoptimization algorithms used in these odes perform poorly for systems witha ompli ated unit ell, su h as zeolites; state-of-art algorithms would bemu h more e� ient. Therefore, a general algorithm that an be used withperiodi programs was developed, and it is des ribed in the �rst hapter.2. The algorithm was used in sear h of stable silver lusters in zeolite Y.Existen e of a linear luster Agn+3 was suggested for this zeolite, hen e wehave tried to on�rm this idea. Equilibrium geometries and relative energies2

of various on�gurations of silver atoms, bearing various total harges, wereinvestigated in the se ond hapter.3. After �nding the silver spe ies, their photo hemi al properties insidezeolites must be determined. Methods that an be e� iently used to modelex ited states of su h large systems�e.g., density fun tional theory (DFT)�are not reliable and have to be ben hmarked against more a urate methodson smaller systems. The last hapter presents a study of opti al propertiesof Agn+3 lusters in a simple model framework of two F � anions. Varioussingle- and multireferen e methods were used to model ele troni ally ex itedstates and they were ompared to DFT-based methods.

3

Chapter 1Geometry optimizationThe omplete quantum-me hani al des ription of a mole ule is en oded inthe wavefun tion depending on both nu lear and ele troni oordinates. TheBorn-Oppenheimer (BO) approximation states that motions of nu lei andele trons an be separated be ause nu lei are mu h heavier than ele trons�the momentum onservation prin iple then implies that ele trons are movingat mu h higher speed and they �see� nu lei motionless at given time. The BOapproximation is also alled the adiabati approximation. This name omesfrom the adiabati theorem whi h states that an eigenstate of a system ispreserved when a small time-dependent perturbation (geometry hange inthis ase) is varying slowly.Upon applying the BO approximation, the mole ular S hrödinger equa-tion splits into two separated equations. The �rst one des ribes the motionof ele trons in the ele trostati �eld of nu lei, and its solutions are ele troni states and their energies. The ele troni energy added to the nu lear repul-sion gives the total energy E, whose dependen e on the positions of nu leiis alled the potential energy surfa e (PES). The se ond equation des ribesthe motion of nu lei in the e�e tive potential given by the PES [6℄.The position of ith nu leus will be denoted Ri. Positions of N nu lei anbe arranged into a 3N-dimensional olumn ve torx =

(

Rx1 Ry

1 Rz1 Rx

2 Ry2 . . . Rz

N

)t (1.1)To des ribe important points of the PES, the gradient g and Hessian H haveto be de�ned,gi =

dE

dxi

Hij =∂2E

∂xi∂xj

(1.2)Points where g = 0 are alled stationary points. Depending on eigenvalues ofH at these points, they an be minima, saddle points or maxima. Eigenvalues4

of Hessian play a role of the se ond derivative in one-dimensional analysis�they are all positive at the minimum, there is just p negative ones at thepth-order saddle point and they are all negative at the maximum.If the system under study is stable, there has to exist at least one lo alminimum. The minima are more or less stable mole ular geometries andare also alled equilibrium geometries. They an orrespond to intermedi-ates or stable ompounds, depending on the shape of the PES around themand on thermodynami onditions. Another distin tive points important for hemists are �rst-order saddle points whi h orrespond to transition statesin hemi al rea tions. The eigenve tor orresponding to the only negativeeigenvalue of H lies in the dire tion of a hemi al rea tion.Almost every study in theoreti al hemistry begins with a sear h of anequilibrium geometry. Be ause of high dimensionality of this problem, it anbe done systemati ally only for smallest mole ules. For bigger systems, the on ept of walking on the PES has to be introdu ed; one starts with someinitial geometry whi h is assumed to be lose to the equilibrium geometry,performs the energy al ulation and evaluates a set of rules ( alled optimiza-tion algorithm) whi h predi ts a step to di�erent geometry, hopefully loserto the minimum. This is repeated until the equilibrium geometry is rea hed.In addition to the total energy of a system, also the gradient and Hessian an be used in the sear h. When they are al ulated expli itly, they are alled analyti . When they are obtained by al ulating the total energy atdispla ed geometries, they are alled numeri al.The only measure of the optimization's e� ien y is omputation time it onsumes. This is given by the ost of one step multiplied by the number ofthe steps. In most methods, the additional analyti evaluation of the gradientdoes not in rease the step ost signi� antly, while its knowledge redu esthe number of steps rapidly. Therefore, it is advantageous to al ulate thegradient expli itly. On the ontrary, this does not hold for the evaluation ofthe Hessian whi h is very expensive ex ept for the simplest methods, hen eit is performed only for spe ial purposes. There are also methods, su h aspopular CCSD(T), for whi h analyti gradients are not implemented in mostprograms. In these ases, the gradient is usually omputed numeri ally.Our optimization algorithm will be used mainly for DFT al ulations.The Kohn-Sham realization of the DFT has available heap gradients but anexpensive Hessian. Hen e we fo used on the lass of optimizers whi h employthe energy and the gradient.A naive idea is to make a step in the dire tion opposite to the gradient�this is alled the steepest des ent method. The state-of-art algorithms use aquadrati approximation of the lo al PES and try to jump in its minimum inone step. This requires the knowledge of the Hessian and as it is not readily5

available, it has to be approximated by taking into a ount all gradients fromprevious steps. To re over the exa t Hessian, one would have to know thegradients in 3N linearly independent points but even the guesses based on afew points fasten the onvergen e rapidly.1.1 MethodsOur algorithm onsists of several independent ideas whi h were taken fromvarious sour es. The main outline omes from the Berny algorithm that wasoriginally des ribed in [7℄. It was implemented in Gaussian, later modi�ed,and it is loosely des ribed in the Gaussian manual urrently. Our implemen-tation of the methods is original and will be do umented in Se tion 1.2.1.1.1 Quadrati approximationThe entral part of the algorithm is a lo al approximation of the PES at thepoint x0 by a quadrati fun tionE(x) = E0 + gt∆x + 1

2∆xtH∆x (1.3)where ∆x = x−x0 and E0, g, H are the energy, gradient and Hessian at x0.Newton-Raphson step A step ∆x that leads into the lo al extreme ofthe quadrati surfa e an be made

∆x = −H−1g (1.4)It is alled the Newton-Raphson (NR) step. As the quadrati expansion isjust an approximation, the NR step does not generally lead to the minimumof the PES. But it lowers the energy when x0 is lose to the minimum.The NR step an fail ompletely under ertain onditions. When the lo alPES has wrong urvature (some Hessian eigenvalues are negative), the NRstep does not have to lead downhill. A tually, it always leads uphill at themaximum.1 The NR step an also be very large and an lead to the regionwhere the quadrati approximation does not hold. This happens when thegeometry is far from a minimum (the gradient has large magnitude), or whenthe PES is very �at (some eigenvalues of Hessian are very small).1However, one rarely deals with maxima of the PES.6

Rational fun tion optimization The quadrati surfa e is modi�ed inthe RFO method [8℄E(x) = E0 +

gt∆x + 12∆xtH∆x

1 + ∆xt∆x(1.5)Sear h for stationary points of this surfa e gives the eigenvalue equation

(

H g

gt 0

)(

∆xi

1

)

= λi

(

∆xi

1

) (1.6)where ∆xi is the predi ted step, λi is twi e the hange of the energy or-responding to this step, and i ranges from 1 to 3N + 1. After sorting theeigenvalues, the �rst being the smallest, the ith step leads to the ith-ordersaddle point.How is this method onne ted with the NR step and why it enables goingdownhill even in regions with a wrong urvature an be seen from its di�erentformulation∆x = −(H − λ)−1g (1.7)This is just a �shifted� NR step, with all Hessian eigenvalues made e�e tivelypositive (λ1 < 0), orre ting the possibly wrong PES urvature.Trust radius The NR and RFO methods use an approximation to the realPES in order to predi t a minimizing step. This approximation an be veryina urate far from the origin, i.e., one an �trust� the approximation just insome region around x0. Therefore, the minimizing step is required to satisfythe ondition

‖∆x‖ < T (1.8)where T is alled the trust radius (TR).If the generated NR step is shorter than T , it is a epted and the step ismade. If it is longer, minimization of the quadrati surfa e on a sphere ofthe radius T has to be performed. This leads to the same equation as theRFO approa h, but one looks for a di�erent solution [6℄. All steps of theform (1.7) with λ < λ1 have di�erent length, and ea h of them minimizesthe energy on a sphere with radius equal to their length. To �nd the stepminimizing the energy on a sphere of the radius T , one has to solve∥

∥−(H− λ)−1g∥

∥ = T (1.9)and put the obtained λ < λ1 ba k into (1.7).The size of T is hanged dynami ally during the optimization. One startswith some initial T (1) and iteratively updates it, based on the ratio ξ of the7

energy hange that a tually o urred after making the step and the energy hange predi ted by the PES approximation [9℄T (i+1) =

14

∥

∥∆x(i)∥

∥ ξ < 0.25

2T (i) ξ > 0.25 and ∥∥∆x(i)∥

∥ = T (i)

T (i) otherwise (1.10)Hessian update It was already mentioned that the Hessian is only ap-proximated. It is done by starting with some initial guess and then updatingthe Hessian in ea h step. There exists a large number of various formulas forthis task; we use a ombination of the BFGS and MS s hemes [10℄H(n+1) = H(n) + φ∆HBFGS + (1− φ)∆HMS (1.11)where∆HBFGS =

∆g∆gt

∆xt∆g− H(n)∆x∆xtH(n)

∆xtH(n)∆x

∆HMS =(∆g −H(n)∆x)(∆g −H(n)∆x)t

∆xt(∆g −H(n)∆x)

φ =

∥

∥

∥∆xt(∆g −H(n)∆x)

∥

∥

∥

‖∆x‖∥

∥

∥∆g −H(n)∆x

∥

∥

∥1.1.2 Line sear hThis method estimates a minimum of the energy on the line onne ting twopoints with known energies and gradients without the additional ost of theenergy al ulation. It is done by exa t �tting of a one-dimensional fun tionF (t) on energies and its derivatives (proje ted gradients) in the two points.To �nd a minimum between the points x0, x1 with the orrespondingenergies E0, E1 and gradients g0, g1, the gradients are �rst proje ted

fi = (x1 − x0)tgi (1.12)Four data points are then used to �t the fun tion F

F (0) = E0 F (1) = E1

F ′(0) = f0 F ′(1) = f1(1.13)This means that four-parameter fun tions F are needed.8

The point minimizing the energy is given byxmin = x0 + tmin(x1 − x0) (1.14)where tmin minimizes F .Constrained quarti polynomial A quarti polynomial ∑4

n=0 cntn has�ve oe� ients. One degree of freedom an be removed by onstraining it insu h a way that it has a zero se ond derivative in one point only. This ensuresthat the polynomial has either one lo al minimum or one lo al maximum,and no other stationary points. Depending on the data points, no su hpolynomial exists or two di�erent polynomials exist that an be �tted. Theone with a lower minimum is always hosen.To �t the onstrained quarti polynomial, the dis riminant D is �rst al ulated.

D = −(f0 + f1)2 − 2f0f1 + 6(E1 −E0)(f0 + f1)− 6(E1 − E0)

2 (1.15)If D < 0, no quarti polynomial an be �tted. If D ≥ 0, the oe� ients oftwo possible polynomials are given byc0 = E0 c1 = f0

c2 =12

(

−5f0 − f1 − 6y0 + 6y1 ±√2D)

c3 = −2c2 − 4(E0 − E1)− 3f0 − f1

c4 = c2 + 3(E0 − E1) + 2f0 + f1

(1.16)Cubi polynomial A ubi polynomial∑3

n=0 cntn an be always �tted onfour data points.

c0 = E0 c1 = f0

c2 = −3(E0 − E1)− 2f0 − f1

c3 = 2(E0 −E1) + f0 + f1

(1.17)1.1.3 Internal redundant oordinatesCartesian oordinates are simplest to implement but ine� ient for geometryoptimizations. A better hoi e is to use non-redundant internal oordinates�a set of bond distan es r, valen e angles θ and dihedral angles ϕ that uniquelydes ribe the mole ule. However, the best hoi e for geometry optimization9

was shown to be internal redundant oordinates [11℄. Due to their redun-dan y, they require a spe ial treatment and some of the aforementionedmethods must be modi�ed.The internal oordinates an be generated automati ally from the Carte-sian oordinates. First, two atoms i and j are regarded as bonded if theirdistan e rij is smaller than 1.3 times the sum of their ovalent radii Cij. Ifthere are two or more fragments that are not onne ted by this riterion, theshortest distan e between the fragments is determined. All interfragmentdistan es less than 1.2 times this distan e or less than 2Å are then desig-nated as bonds. Next, all possible valen e angles and dihedral angles aregenerated.The internal oordinates q are al ulated by following formulasrij = ‖vij‖

θijk = arccosvtijvkj

‖vij‖ ‖vkj‖

ϕijkl = sgn det(

vlk vij vkj

)

arccosatijalk

‖aij‖ ‖alk‖

(1.18)whereaαβ = vαβ − (vt

αβekj)ekj ekj = vkj/ ‖vkj‖vαβ = Rα −Rβ

(1.19)Weighing the oordinates The optimization of non- ovalent, weak bonds(realized by van der Waals for es) onverges very slowly. This is aused bygradients and Hessian eigenvalues being small in their dire tions. To ir- umvent this problem, the oordinates an be weighted a ording to theirstrength [12℄.The degree of ovalen e of a bond between atoms i and j is de�ned asρij = exp

(

− rijCij

− 1

) (1.20)The weight of the nth oordinate is then al ulated aswn =

ρij bond rij√ρijρjk[f + (1− f) sin θijk] angle θijk

3√ρijρjkρkl

∏

α∈{ijk,jkl}[f + (1− f) sin θα] dihedral ϕijkl

(1.21)where f = 0.12. The weights are arranged into the diagonal weight matrixw. 10

The numbers ρij are also used for estimating the initial diagonal Hessian.Hmm =

0.45ρij bond rij

0.15ρijρjk angle θijk0.005ρijρjkρkl dihedral ϕijkl

(1.22)The Hessian is thus reated in the internal oordinates q, using the weight-s aling (w), and it is maintained like this during the whole al ulation.Therefore, the Hessian will be denoted Hw,q from now on.Wilson B matrix The gradient and Hessian have to be transformed fromthe Cartesian oordinates x to the internal oordinates q prior to the use ofquadrati methods. This is done by the B matrix de�ned asBmn =

∂qm∂xn

(1.23)The parti ular analyti expressions were obtained by di�erentiating (1.18).For bond distan es, they are∂rij∂Ri

=vij

‖vij‖∂rij∂Rj

= − vij

‖vij‖(1.24)and for valen e angles

∂θijk∂Ri

=vij cotg θijk

‖vij‖2− vkj

‖vij‖ ‖vkj‖ sin θijk∂θijk∂Rj

=vij + vkj

‖vij‖ ‖vkj‖ sin θijk−(

vij

‖vij‖2+

vkj

‖vkj‖2

)

cotg θijk

∂θijk∂Rk

=vkj cotg θijk

‖vkj‖2− vij

‖vij‖ ‖vkj‖ sin θijk

(1.25)For dihedral angles, three di�erent ases have to be al ulated: a generalangle ϕ, and limiting ases ϕ = 0 and ϕ = π. For a general angle, theformulas are∂ϕijkl

∂Ri

=aij cotgϕijkl

‖aij‖2− alk

‖aij‖ ‖alk‖ sinϕijkl

∂ϕijkl

∂Rj

=(1− A)alk − Baij

‖aij‖ ‖alk‖ sinϕijkl

−(

(1− A)aij

‖aij‖2− Balk

‖alk‖2

)

cotgϕijkl

∂ϕijkl

∂Rk

=(1 +B)aij + Aa3

‖aij‖ ‖alk‖ sinϕijkl

−(

(1 +B)alk

‖alk‖2+

Aaij

‖aij‖2

)

cotgϕijkl

∂ϕijkl

∂Rl

=alk cotgϕijkl

‖alk‖2− aij

‖aij‖ ‖alk‖ sinϕijkl

(1.26)11

whereA = vt

ijekj/ ‖vkj‖ B = vtlkekj/ ‖vkj‖ (1.27)For the spe ial ases, we �rst de�ne

u =vkj × aij

‖vkj × aij‖(1.28)and the formulas are

∂ϕijkl

∂Ri

= u/ ‖aij‖

∂ϕijkl

∂Rj

=

(

−1 −A

‖aij‖∓ B

‖alk‖

)

u

∂ϕijkl

∂Rk

=

(

±1 +B

‖alk‖− A

‖aij‖

)

u

∂ϕijkl

∂Rl

= ∓u/ ‖alk‖ .

(1.29)where the upper signs +,− hold for ϕ = 0 and the lower for ϕ = π.Transformation While the gradient g obtained from quantum hemi alprograms is usually in Cartesian oordinates, the quadrati optimization isperformed in the non-redundant subspa e of the internal redundant oordi-nates with a weight-s aled gradient and Hessian. This paragraph des ribestransformation between these two oordinate systems.First, the G matrix is al ulated

G = BBt (1.30)This matrix is singular be ause of the redundan y of the used oordinates.The generalized inverse G− is obtained by inverting only the nonzero eigen-values in the basis of the eigenve tors V.VtGV =

(

E 0

0 0

)

G− = V

(

E−1 0

0 0

)

Vt (1.31)The weight-s aled gradient in internal oordinates is then obtained bygw,q = w−1G−Bg. (1.32)Next, a proje tor on the non-redundant subspa e is al ulated

P = GG− (1.33)12

The proje ted weight-s aled gradient gw,proj and HessianHw,proj are obtainedbygw,proj = Pgw,q Hw,proj = PHw,qP+ γ(1−P) (1.34)where γ = 1000 au. These quantities are then used in the RFO method.The step ∆q generated by the RFO method is in internal oordinates andmust be onverted ba k to Cartesian oordinates. This onversion is deter-mined by non-linear equations (1.18), therefore, the pro edure is performediteratively in linearized (hen e the use of B matrix) oordinates2

∆x(i) = BtG−∆q(i) x(i+1) = x(i) +∆x(i) (1.35)The new internal oordinates q(i+1) are then al ulated from the obtainedx(i+1) in ea h iteration, their a tual hange is ompared to the intended one

∆q(i+1) = ∆q(i) − (q(i+1) − q(i)) (1.36)and the new step ∆q(i+1) is inserted into (1.35) again. The iteration isstopped after the kth step when rms(∆x(k)) < 10−6 au. Finally, the Cartesianstep ∆x whi h orresponds to the internal step ∆q is given by x(k) − x(0).1.2 AlgorithmThe program works iteratively. First, a quantum- hemi al al ulation isperformed with an initial geometry and the resulting energy and gradientare saved. Then the geometry optimizing ode is alled. It loads the energyand gradient, runs the algorithm and generates the next geometry. Thispro edure is then repeated until the onvergen e is rea hed. The �rst andnext runs of the algorithm di�er signi� antly and will be des ribed separately.First run1. Load the initial geometry x(1) given in Cartesian oordinates.2. Generate the internal oordinates a ording to (1.18).3. Cal ulate the weights of the oordinates (1.21).4. Estimate the initial Hessian guess given by (1.22).5. Set the initial trust radius to 0.3 au.6. Load the energy and gradient for the initial geometry.7. Cal ulate the Wilson B matrix (1.23) and the proje tor on the non-redundant subspa e (1.33).2In this paragraph, the upper index (i) denotes this (inner) iterative y le, not the(outer) geometry optimization y le. 13

8. Constru t the RFO matrix (1.6) using the proje ted weight-s aled gra-dient and Hessian (1.34) an obtain the quadrati step ∆q(1). If thestep is shorter than the trust radius, a ept it and go to the next step.Otherwise(a) solve (1.9) for a step with an a eptable length.9. Che k for the onvergen e riteria (1.37) and eventually stop.10. Convert the geometry step from internal to Cartesian oordinates asdes ribed by (1.35).11. Save the new geometry x(2) and all for the energy�gradient al ulation.nth run1. Cal ulate the Wilson B matrix (1.23) and the proje tor on the non-redundant subspa e (1.33) for the urrent geometry x(n).2. Load the energy and gradient for the urrent geometry and transformthe gradient into internal oordinates (1.32).3. Update the Hessian a ording to (1.11) using the gradient g(n)

w,q and step∆q(n−1).4. Update the trust radius (1.10).5. Perform the linear sear h (1.14) with the points q(n) (t = 0) and q(n−1)(t = 1):(a) �rst, try a quarti polynomial (1.16). If the �t exists and has aminimum, a ept it. Otherwise(b) try a ubi polynomial (1.17). If it has a minimum between thetwo points, a ept it. Otherwise( ) perform no linear step (stay in q(n)) if En < En−1.(d) If En > En−1, perform a linear step ba k to the previous geometry

q(n−1) (t = 1).6. Set the linear step ∆q(n)lin and linearly interpolate the gradient.7. Proje t the Hessian and interpolated gradient (1.34) and use them to onstru t the RFO matrix (1.6).8. Obtain the quadrati step ∆q

(n)quad. If the step is shorter than the trustradius, a ept it and go to the next step. Otherwise(a) solve (1.9) for the step with a eptable length.9. Cal ulate the total step ∆q(n) = ∆q

(n)lin +∆q

(n)quad.10. Che k for the onvergen e riteria (1.37) and eventually stop.11. Convert the geometry step∆q(n) from internal to Cartesian oordinatesas des ribed by (1.35).12. Save the new geometry x(n+1) and all for the energy�gradient al ula-tion. 14

Convergen e riteria The iterative sear h for an equilibrium must bestopped at some point. We use the riteria similar to those used in Gaussian.max(g) < 0.45×10−3 rms(g) < 0.3×10−3

max(∆q) < 1.8×10−3 rms(∆q) < 1.3×10−3 (1.37)The energies and distan es in expressions above are given in atomi unitsand the angles in radians.1.3 ImplementationThe algorithm was implemented in GNUO tave whi h is a high-level pro-gramming language that is interpreted on-the-�y rather than ompiled intoa binary ode. It is intended for numeri al omputation and ontains a largenumber of mathemati al fun tions available as ready-to-use libraries [13℄.The sour e ode of the program is listed in Appendix A. It onsists ofseveral �les, ea h one ontaining typi ally one fun tion. The main�le of theprogram is berny.m. Thanks to a high level of the used language, the main�le ontains a omplete logi al stru ture of the algorithm as des ribed in Se tion1.3, while the other �les ontain only helper fun tions.The program is started by alling the fun tion berny. During the �rstrun, the program reads ontents of the �les e (energy), g (gradient) andxyz (Cartesian geometry) whi h have to be provided by higher layers of the omputational work�ow. After performing the algorithm, the program savesthe generated geometry into the �le xyz and the whole variable environmentinto the �le workspa e.mat. In subsequent runs, it loads the energy, gradientand variable environment.Following helper fun tions were programmed: genind generates the def-inition of internal oordinates, while internals a tually al ulates them;weights returns the weight matrix w; hessian generates the initial Hessianguess; Bmat al ulates the B matrix; fitquarti and fit ubi �t quarti and ubi polynomials, respe tively, given fun tion values and derivatives intwo points. This set of routines provides main fun tionality.The remaining fun tions perform only simple tasks: ginv omputes thegeneralized inverse of a matrix; onn broadens the onne tivity matrix (twoatoms are onne ted if there is a hain that onne ts them); radius returnsthe ovalent radius of an atom; dist omputes the distan e matrix of a setof atoms; angle returns the valen e angle; orre t orre ts angles runningout of the interval (0, 2π); rms returns the root mean square of a ve tor; andwriteX, readX and fileread provide �le I/O operations.15

A layer that provides ommuni ation of the program with the quantum- hemi al pa kages had also to be implemented. As this layer does not performany mathemati al operations, Linux shell s ripts were used onveniently. Ourprogram was linked to the popular programs Gaussian, Turbomole and peri-odi VASP.The program generates a log�le during its run. A typi al example of thelog is presented below.Gaussian terminated normallyEntering O tave al ulation: 99 Energy: −308.390550399 Hessian update information:9 * Bo�ll's oe� ient: 0.6861079 * Maximum absolute hange: 0.0008492949 * RMS hange: 2.38024e−059 Flet her's parameter: −0.1860789 Linear interpolation:9 * Energies: −308.390551, −308.3905509 * Derivatives: −3.29848e−06, 4.02318e−069 * Quarti interpolation was performed: t=−0.585969 Predi ted energy hange from linear step: −1.21432e−069 Trust radius: 0.005382279 RMS of quadrati step: 0.0003988569 Predi ted energy hange from quadrati step: −5.05748e−079 Angle di�eren e to last step: 106.3829 Convergen e riteria:9 * Gradient maximum is: 5.14123e−05, threshold: 0.00045, OK9 * Gradient RMS is: 6.02716e−06, threshold: 0.0003, OK9 * Step maximum is: 0.0034006, threshold: 0.0018, no9 * Step RMS is: 0.000469583, threshold: 0.0012, OK9 Iteration #2: rms(dx)=5.31102e−07, rms(dq)=0.000609337O tave ended normallyEntering Gaussian al ulation: 10TestsThe algorithm was tested and ben hmarked on a small-sized van der Waalssystem. It onsists of one water mole ule weakly bonded to a benzenemole ule and it is depi ted in Figure 1.1. The distan e between the entreof the aromati ring and the oxygen atom will be referred to as the bond-ing distan e R whi h is equal to 4.0Å in the initial testing geometry. Theequilibrium bonding distan e of the system in DFT/PBE model is ∼3.4Å,depending on many other parameters of the method whi h are not importantin following dis ussion.The results presented in Table 1.1 show omparison of our optimizer with16

Figure 1.1. Mole ular system used in tests andben hmarking of the geometry optimization al-gorithm.Table 1.1. Comparison of geometry optimizations per-formed by our program and the default algorithms in Gaus-sian 09 and VASP.algorithm Na Rb ∆E rms(g)dGaussian 09 default 18 3.355 −2.8602 3.0our work 10 3.354 −2.8602 1.8VASP default 13 3.993 −1.1734 >1034 3.981 −1.1904 >10100 3.959 −1.2392 >10our work 13 3.499 −1.8036 1.7a Number of energy al ulations needed to rea h the givengeometry.b Bonding distan e in angstroms. Energy hange during the optimization in mH.d Gradient of the energy times 104 in atomi units.the ones provided in Gaussian 09 and VASP. It an be seen that our al-gorithm performs almost two times better than that in Gaussian whi h is onsidered to be among the best of ommer ial programs; the �nal bondingdistan e and energy are in perfe t agreement. The di�eren e in performan eis enormous in ase of VASP. While our algorithm onverged in 13 steps,VASP was no near to the equilibrium geometry even after 100 steps: thebonding distan e shortened only by 0.04Å. The VASP driver probably opti-mized only the strong ovalent bonds, but it annot optimize also the weakones.As for speed of the program itself, its omputation time is negligible in ase of systems with less then 50 atoms. When testing on bigger systems,it was found out that the bottlene k are two diagonalizations of Hessian-17

sized matri es performed in ea h step. This ould be possibly improved bydiagonalizing the matri es outside of O tave. But it is not very important forthe overall performan e of the optimization be ause the energy al ulationtakes still mu h more omputational time.

18

Chapter 2Silver lusters in zeolite YA key step in understanding of Ag/zeolite�based �uores ent emitters is theinvestigation of the stru ture and distribution of silver lusters formed in-side zeolite hannels and avities. As was mentioned in the introdu tion, asuggested explanation for their formation is redu tion of silver. Hen e, the lusters with di�erent harges have to be investigated.Many di�erent zeolites have been probed as hosts for silver lusters.Among the most studied, faujasite (FAU) was hosen for our study. It hasa topology shown in Figure 2.1. The spa e group of the rystal is Fd3̄m,whi h has a fa e entered ubi latti e (a ≈ 25.0Å, depends on the Si/Alratio and extraframework parti les), and belongs to the point group Oh [14℄.The unit ell onsists of eight sodalite ages (SOD) and ontains 576 frame-work atoms. This unit ell is too large to be reasonably modelled by ab-initiomethods, but a redu ed unit ell exists fortunately. The redu tion is possibledue to existen e of the subgroup R3̄m of the rystal spa e group. This spa esubgroup has a rhombohedral latti e (a ≈ 17.4Å, α = 60°) and belongs tothe point group D3d. The redu ed unit ell ontains only two sodalite agesand is depi ted in Figure 2.2.Zeolites with the same topology an di�er in the Si/Al ratio and thedistrubution of Al atoms. Two lasses of faujasite are distinguished a ord-ing to whether their Si/Al ratio is lower or higher than 1.5: the formerare named zeolites X, the latter zeolites Y. Among the Si/Al ratios stud-ied experimentally [3℄, zeolite Y with Si/Al of 2.7 was hosen in our study.It was realized by having 35 Si atoms and 13 Al atoms in the unit ell.Aluminium atoms were distributed randomly while respe ting the followingrules:1 (i) Löwenstein's rule stating that Al−O−Al hains never o ur in ze-olites, (ii) Dempsey's rule stating that the distribution of Al atoms minimizes1I would like to thank Luká² Graj iar for reating a program that an generate su hdistributions. 19

Figure 2.1. Faujasite topology onsisting of red sodalite ages and bluedouble-6-rings. Si and Al atoms are lo ated at the verti es and oxygenatoms lie on the edges. The unit ell from Figure 2.3 an be seen on theright side of this �gure.the number of Al−O−Si−O−Al hains and (iii) the rule that populations ofSiO4 tetrahedra adja ent to 1, 2, 3 or 4 AlO �4 tetrahedra have to orrespondto the experimental ones obtained by 29Si NMR.Cation distributionNa+ was hosen as a ompensating ation. Sin e the unit ell frameworkbears a total negative harge of 13 AlO �4 tetrahedra, there must be 13 Na+ ations. It is known that ompensating ations o upy only several distin-guished positions in the zeolite stru ture alled extra-framework ation sites(see Figure 2.2).Site I is lo ated in the middle of a double-6-ring (D6R).Site I′ is in the middle of a 6-ring forming an entran e to a D6R and slightlydispla ed into the SOD.Site II is in the middle of a 6-ring that forms a SOD and slightly displa edout of the SOD into the super age.There exist a tually more sites in faujasite topology, but they are not relevantfor this work.There are 4 sites I, 8 sites I′ and 8 sites II in our unit ell. The number ofpossibilities how to o upy these 20 sites by 13 ations is large. Fortunately,it an be redu ed greatly by following reasoning. It is known from experimentthat the relative stability of sodium ations in di�erent sites de reases in orderII > I > I′ in faujasite [15℄. Therefore, the most stable distribution of 12 Na+ ations would be 8 sites II and 4 sites I. As there are no more sites II or I leftfor the 13th ation, it has to o upy site I′. However, as Na+ ations annot20

Figure 2.2. Unit ell used in al- ulations. It is rhombohedral (a =17.372Å, α = 60°) and ontains 35 Siatoms (grey), 13 Al atoms (pink) and96 oxygen atoms (red). Three ationpositions (violet) are shown: site I, I′and II.

Figure 2.3. Geometri representa-tion of the unit ell. It onsists oftwo sodalite ages (bottom is �1�, topis �2�) and four double-6-rings (D6R)labeled A,B,C,D. Eight 6-rings arepresent (a1, a2, . . . , d1, d2). The greenspots spe ify oxygen �1� in ea h D6Rand the arrows the dire tion of num-bering. Detailed des ription of the no-tation is in the text.o upy adja ent sites I and I′ at the same time, one ation from the site I hasto be moved into the site I′ additionally. The resulting distribution is thus8 sites II, 3 sites I and 2 sites I′. This is already unique from a topologi alview, however, the presen e of Al atoms makes sites of the same kind infa t inequivalent. E.g, the four D6R's di�er in the number of ontained Alatoms in our parti ular Al distribution. Therefore, the aforementioned Na+distribution omprises four Na+ on�gurations di�ering in whi h site I is�split� into two sites I′. To uniquely des ribe these on�gurations, we havedeveloped a notation des ribed below.NotationThe notation is introdu ed in Figure 2.3. This �gure is a simpli�ed geometri representation of the unit ell from Figure 2.2. The bottom and top SOD's21

are labeled �1� and �2� respe tively. Four D6R's in the unit ell are labeledA,B,C,D as shown in the pi ture. The orresponding sites I bear the samename. Sites I′ are labeled as A1, A2, . . . , D1, D2, denoting the parti ular SOD(subs ript) and parti ular D6R. Finally, sites II are expli itly named in the�gure as a1, a2, . . . , d1, d2, denoting the parti ular SOD and D6R (in lower ase) to whi h they are opposite. The positions in Figure 2.2 are thus C (I),D2 (I′) and c2 (II).Returning to the question of whi h site I should be split, the D6R's labeledA and B ontain 4 Al atoms (denoted 4Al D6R), C three and D just two.As the split D6R has to ompensate for one additional ation, a reasonableassumption is that this D6R should bear the greatest negative harge�itshould be A or B. Let it be A for a moment. The resulting on�guration ofsilver atoms is then written as8II BCD A1A2The re ord onsists of three blo ks des ribing sites II, I and I′ respe tively.8II means that all eight sites II are o upied. BCD means that sites I labeledB, C and D are o upied, while the remaining A is uno upied. The lastblo k has the same meaning for sites I′.If, say, the ation from the site II labeled a1 was removed, the re ordwould read 7II(−a1) BCD A1A2Now the �rst blo k means that seven sites II are o upied and the one thatis missing to full o upation is a1. All on�gurations en ountered later inthe text an be des ribed by this system.Silver ationsSome of the Na+ ations an be ex hanged for Ag+. Full range from zero tothirteen ex hanged ations an be rea hed, however, to keep things as simpleas possible, we fo used only on two possibilities in our study. (i) When onlyone ation is ex hanged, the resulting on�gurations an be ompared tosee whi h sites are favourable for silver ations. (ii) When three ations areex hanged, stru ture of Agn+3 lusters an be investigated. An importantexample is a trigonal luster in a D6R (I′−I−I′) be ause it was suggestedbased on experiment, that Ag+ an o upy adja ent sites I and I′ at thesame time [16℄. Other geometries and positions an also be investigated:linear and non-linear lusters in a SOD, or ompletely isolated silver ations.The harge of the lusters an be experimentally ontrolled by redu tion.To redu e the silver lusters, a hydrogen atom was introdu ed into the unit22

ell. Its ele tron redu es the luster, while the resulting H+ ation is bondedto an oxygen atom of the zeolite framework. The most stable positions forH+ are the D6R oxygen atoms neighbouring Al atoms [17℄. These are namedas A1, . . . , A6, B1, . . . , D6, where the letter denotes a parti ular D6R andthe numbering is explained in Figure 2.3 (see �gure aption). Two di�erentredu ed silver lusters were studied: Ag 2+3 and Ag+3 , where the appropriatenumber of hydrogen atoms needed to a hieve this harge is one and tworespe tively.MethodsThe relative energies of various Na/Ag on�gurations were investigated byperiodi DFT al ulations as implemented in VASP [18, 19℄, using the Per-dew�Burke�Ernzerhof (PBE) fun tional [20℄. The proje tor augmented-wave(PAW) method was employed [21, 22℄. Standard PAW pseudopotentials forSi, Al, O, H, Na and Ag were used with the enmax values of 245, 240, 400,250, 102 and 250 eV respe tively. The plane wave basis set with an energy ut-o� of 400 eV was used. The k-points sampling was restri ted to theΓ-point. The onvergen e riteria for the ele troni energy was set to 10−6 eV.The onvergen e riteria for geometry were des ribed in Chapter 1 by Eq.1.37.2.1 ResultsThe equilibrium geometries and energies of various sodium and silver on�g-urations are summarized in four tables. Con�gurations ontaining no silveror one silver atom are in Table 2.1. Non-redu ed on�gurations with threesilver ations are in Table 2.2. Con�gurations redu ed with one and twohydrogen atoms are in Tables 2.3 and 2.4, respe tively.Index n running over all four tables is used to onveniently refer to the on�gurations from the text. If a distan e between silver atoms is shorterthan 4Å, these atoms are en losed in bra kets (denoting a luster) and theirgeometry is reported. A parti ular meaning of ea h geometry re ord is ex-plained in the table footnotes.Relative stabilities of di�erent on�gurations of 13 Na+ ations in theunit ell are reported in the upper part of Table 2.1. While a D6R with 4 Alatoms (A) is split in on�guration 1, a 2Al D6R (D) is split in on�guration2. It an be seen that the D6R with more Al atoms is preferred (−30 kJ/mol).Relative stabilities of one ex hanged silver ation in various sites are pre-sented in the lower part of Table 2.1. It an be seen that ex hange of the23

Table 2.1. Relative equilibrium energiesof extra-framework ation on�gurationsin zeolite Y: Na+13 and Ag+Na+12.Con�gurationn Silver Na+ Ea1 � 8II BCD A1A2 0.02 � 8II ABC D1D2 30.33 B 8II CD A1A2 0.04 D 8II BC A1A2 1.45 C 8II BD A1A2 1.66 b1 7II(−b1) BCD A1A2 11.87 A1 8II BCD A2 13.98 d1 7II(−d1) BCD A1A2 15.5a Relative energy in kJ/mol. ation in site I is most favourable and that the di�eren e oming from thenumber of Al atoms in the D6R is almost negligible (4Al is 1 kJ/mol below3Al and 2Al).Ex hange of the ation in sites I′ and II (both are in the middle of a6-ring) is more than 10 kJ/mol higher than that of the ation in site I. In ontrast to a D6R, the stability depends on the number of Al atoms in a6-ring�b1 ontains two Al atoms while d1 only one, and this fa t makes adi�eren e of 4 kJ/mol.Three silver ations, Ag 3+3Con�gurations reported in Table 2.2 ontain three silver ations. Five moststable on�gurations have silver ations far from ea h other so they annotintera t. The most stable of them (9) has all silver ations in sites I and itserves as a referen e on�guration. All other four (10�13) are ∼30 kJ/molhigher in energy. This is in agreement with the results from Table 2.1�e.g., on�guration 10 has two silver ations in sites I′ instead of in sites I, and thisis expe ted to be (see on�guration 7 vs. 3) 2 × 14 = 28 kJ/mol above thereferen e, and it is a tually al ulated to be 26 kJ/mol.The proposed linear luster in a D6R is investigated in the remaining on�gurations (14�21). Even the most stable ase (14, luster in D6RA) ismore than 100 kJ/mol above the referen e. The average distan e betweensilver atoms in these lusters is about 3.2Å. The e�e t of the number of Alatoms in a D6R is even more pronoun ed here� omparing 14 with 18 and15 with 19, it an be seen that the linear luster is by 70 kJ/mol more stable24

Table 2.2. Relative equilibrium energies of extra-framework ation on�gurations in zeolite Y: Ag+3 Na+10.Con�gurationn Silvera Na+ Geometrya Eb9 BCD 8II A1A2 � 0.010 A1A2B 8II CD � 26.211 b1Ab2 6II(−b1b2) CD B1B2 � 27.512 ABC 8II D1D2 � 31.313 b1Db2 6II(−b1b2) BC A1A2 � 32.014 [A1AA2]

7II(−d2) BCD 3.23, 3.24 107.415 [A1AA2] 8II BC 3.21, 3.31 126.516 a2[DD1] 6II(−a2b1) BC A1A2 3.04 136.717 [A1AA2] 6II(−b1b2) CD B1B2 3.03, 3.30 175.618 [D1DD2] 7II(−b1) ABC 3.19, 3.22 178.319 [C1CC2] 8II BD 3.09, 3.30 198.220 [D1DD2] 6II(−b1b2) BC A1A2 3.01, 3.05 224.821 [A1AA2] a1a2d1d2 B1B2C1C2D1D2 2.87, 2.82 249.6a If a distan e between two silver atoms is greater than 4Å, theyare en losed in bra kets (denoting luster) and their distan e inangstroms is reported.b Relative energy in kJ/mol. Linear luster in a D6R.in a 4Al D6R than in a 3Al or 2Al D6R.When inspe ted in detail�e.g., ompare 14 and 15 with 8 and 4�it anbe seen that the overall harge distribution plays an important role. Afterforming a luster in a D6R ( on entrating the harge), the relative stabilityof di�erent sites hanges. This means that statements su h as ex hange of ations in site I is preferable to ex hange of ations in site II is valid onlyfor the given on�guration of remaining ations, and that su h observations annot be simply generalized.One hydrogen atom, Ag 2+3All on�gurations in Table 2.3 are redu ed with one hydrogen atom. Itsposition is uniquely des ribed (as explained above) but only the fa t thatseveral di�erent H+ positions were tried a tually matters in the followingdis ussion. The same referen e on�guration (33) is used as in the non-redu ed ase ( on�guration 9) ex ept for the additional hydrogen ation.Con�gurations 22�30 are nonlinear Ag 2+3 lusters positioned in a SOD.They posses D3h or C2v symmetry and the Ag�Ag distan e is about 2.8Å.Two most stable on�gurations (140 kJ/mol above the referen e) are of D2h25

Table 2.3. Relative equilibrium energies of Ag 2+3 on�gurations inNa10Ag3HY. Con�gurationSilvera Na+ H+ Geometrya Eb22 [a2B2D2] 7II(−a2) ABC D2 2.77�2.88 −149.823 [B2C2D2] 7II(−a2) ABC D2 2.79�2.84 −141.624 [a2A2B2]d 7II(−a2) ABC D2 2.74, 2.82, 98° −117.125 [a2A2B2]d 7II(−a2) BCD D2 2.82, 2.84, 97° −116.526 [A2B2D2] 7II(−d2) ABC D2 2.78�2.99 −82.827 [a2A2B2]d 7II(−a2) BCD A6 2.72, 2.78, 95° −68.228 [A2B2D2] 7II(−d2) ABC A6 2.75�2.90 −56.929 [A2B2D2] 7II(−a2) ABC A6 2.77�2.92 −53.130 [a2A2B2]d 7II(−a2) ABC Ag3e 2.72, 2.79, 87° −38.831 AB O(A6)f 8II CD Ag−O 2.22, 1.61, 175° −9.532 [A1AA2]g 8II BC D2 3.38, 3.34 −5.333 BCD 8II A1A2 D2 � 0.034 AC O(A6)f 7II(−b2) DB1B2 Ag−O 2.23, 1.61, 177° 5.735 [A1AA2]g 7II(−d2) BCD D2 3.26, 3.66 10.136 d2[AA1] 7II(−d2) BCD A6 3.95 16.337 BCD 8II A1A2 B1 � 17.638 b1A

h 6II(−b1b2) CD B1B2 A6 � 22.639 b1Dh 6II(−b1b2) BC A1A2 A6 � 44.040 [A1AA2]

g 8II BC Ag(A1) 3.21, 3.47 82.141 [A1AA2]g 8II BC A6 3.59, 3.46 87.842 ABC 8II D1D2 D2 � 88.543 ABC 8II D1D2 A6 � 92.844 BCD 8II A1A2 A6 � 96.0a If a distan e between two silver atoms is greater than 4Å, they are en losedin bra kets (denoting luster) and their geometry is reported. Parti ularmeaning of the geometry re ord is explained in footnotes. Distan es are inangstroms.b Relative energy in kJ/mol. D3h trimer in SOD. Silver atoms are positioned above the stated sites.Geometry indi ates range of Ag�Ag distan es.d C2v trimer in SOD. Geometry states lengths of two Ag�Ag bonds and anglebetween them.e Hydrogen is positioned in the middle of the longest edge of Ag3 triangle. Itsdistan es to Ag atoms are 1.83�1.92Å.f One silver atom is bonded to oxygen in D6R from external side. Thehydrogen atom is bonded to this silver atom. Geometry spe i�es O�Agdistan e, Ag�H distan e and O�Ag�H angle respe tively.g Linear trimer in D6R. Geometry indi ates two Ag�Ag distan es.h The third silver atom is on external side of SOD. No atom is in its bondingdistan e. 26

symmetry�the overlap of 5s orbitals on silver atoms is stronger than in theC2v symmetry, and the bonding energy is thus greater. Con�gurations withthe D2 hydrogen are more stable than those with the A6 hydrogen. Theleast stable of luster-in-SOD on�gurations (30) has hydrogen bonded notto an oxygen atom but to the silver luster. The destabilization oming fromthis is apparent, though not ru ial ( ompare with 27). The initial to-be-optimized geometries of some of on�gurations 22�30 were linear lusters inthe SOD. However, they distorted to the �nal non-linear lusters, suggestingthat redu ed linear silver lusters are not stable in a SOD of faujasite.The redu ed linear luster in a D6R is found in on�gurations 32, 35�36and 41�42. The most stable luster of this type is omparable in energy tothe referen e (33). This means that the redu tion by one ele tron stabilizesthe linear luster in a D6R by ∼100 kJ/mol (see on�gurations 9 and 14 for omparison). However, the Ag�Ag distan es are longer than in a non-redu ed ase whi h is in ontradi tion to the expe ted formation of an Ag−Ag bond.Also, one of the Ag�Ag distan es is signi� antly longer than the other one.All this suggests that these on�gurations are one isolated neutral silver atomand two ations rather than silver lusters.Con�gurations 33, 37 and 42�44 have all three silver atoms in sites I. Aninteresting fa t omes from omparing stru tures 33 and 44 with 42 and 43.In the �rst ase, ex hange of the A6 hydrogen for the D2 hydrogen leads to−100 kJ/mol stabilization while it is just −5 kJ/mol in the se ond ase. Thison e again points to the importan e of the overall harge distribution. To beable to dedu e how exa tly is it important, mu h more on�gurations andthorough analysis of various ele trostati ontributions would be needed.Finally, on�gurations 38�39 have one silver atom in the external hannelsystem of the zeoliti stru ture and are 25�50 kJ/mol above the referen e.Two hydrogen atoms, Ag+3In this se tion, the referen e is on�guration 49 whi h is again equivalent to on�guration 9 with two additional hydrogen atoms. The �rst two on�gu-rations in Table 2.4 (45�46) are again the D3h trimers in a SOD. They areboth more than 300 kJ/mol below the referen e and the Ag�Ag distan es areabout 2.7Å (0.1Å shorter than in Ag 2+3 ). The stabilization energy is abouttwo times greater than that for Ag 2+3 whi h roughly orrelates with a fa tthat now there are two 5s ele trons instead of one. Con�guration 47 is asilver dimer in a SOD.Con�guration 48 is again the linear stru ture O−Ag−H, the oxygen atombeing in a D6R. Con�gurations 49�50 are non-intera ting silver atoms, 49being the referen e. Con�gurations 51�52 are linear lusters in a D6R. Their27

Table 2.4. Relative equilibrium energies of Ag+3 on�gurations inNa10Ag3H2Y. Con�gurationn Silvera Na+ H+ Geometrya Eb45 [A2B2C2]

7II(−a2) BCD D2A3 2.71�2.74 −353.546 [a2A2B2] 7II(−a2) BCD A6A3 2.66�2.73 −310.547 A[A1C1]d 6II(−b1b2) CD B1B2 A6A3 2.61, 4.17 −201.648 b1A O(A6)e 6II(−b1b2) CD B1B2 C6 Ag−O 2.31, 1.63, 178° −176.549 BCD 8II A1A2 D2A3 � 0.050 b1Ab2 6II(−b1b2) CD B1B2 A6A3 � 7.351 [A1AA2]f 7II(−d2) BCD A3A6 2.80, 2.83 14.452 [A1AA2]f 7II(−d2) BCD A6C6 2.81, 2.84 22.6a If a distan e between two silver atoms is greater than 4Å, they are en losed inbra kets (denoting luster) and their geometry is reported. Parti ular meaningof the geometry re ord is explained in footnotes. Distan es are in angstroms.b Relative energy in kJ/mol. D3h trimer in SOD. Silver atoms are positioned above the stated sites.Geometry indi ates range of Ag�Ag distan es.d Silver dimer in SOD, its bond length is the �rst Geometry number. The se ondnumber is the distan e between the third silver atom in A and the dimer silveratom in A1.e One silver atom is bonded to oxygen in D6R from external side. The hydrogenatom is bonded to this silver atom. Geometry spe i�es O�Ag distan e, Ag�Hdistan e and O�Ag�H angle.f Linear trimer in D6R. Geometry indi ates two Ag�Ag distan es.Ag�Ag distan e is omparable to the trigonal Ag 2+3 luster but their energiesare a bit above the referen e. This again suggests that there is no bondingbetween silver atoms in the redu ed linear luster in a D6R.2.2 SummaryExisten e of a linear silver luster in a double-6-ring was suggested in theliterature for faujasite. In this hapter, relative energies and geometries ofsilver lusters of various geometries and harges were studied with a on lu-sion that neither a non-redu ed, nor a redu ed linear luster is formed infaujasite in ase of low silver loading. Instead, the non-redu ed silver ationsare lo ated in sites I where they do not intera t with ea h other, and afterredu tion, trigonal lusters are formed inside sodalite ages. The stabiliza-tion of the trigonal lusters seems to roughly orrelate with the number of

5s ele trons, being approximately 150 kJ/mol per ele tron.It has to be emphasized that this study was signi� antly restri ted. The28

results over only (i) Si/Al ratio of 2.7, (ii) one parti ular Al distribution(random but reasonable), (iii) a ompensating ation being Na+ and (iv) alow silver loading (Ag : Na is 3 : 10). However, only the last restri tion isexpe ted to have an e�e t on the qualitative on lusions of the study (one an easily imagine that linear silver lusters would exist in ase of full silverloading, simply be ause there would be no other positions available for silver ations). Nevertheless, �uores en e of the Ag-doped zeolite Y was observedeven for samples with low silver loading [3℄. Therefore, results presented inthis hapter lead to the on lusion that �uores en e annot o ur due to thelinear Agn+3 lusters in Ag-Y with low silver loadings.

29

Chapter 3Ex ited states of silver lustersThe ele troni stru ture of a zeolite an be regarded as a superposition ofthe ele troni stru ture of its framework, the ompensating ations and theguest mole ules. The valen e band of a zeolite framework onsists mainlyof oxygen 2p lone pairs that are lo ated at −10.7 eV, and this is above thevalen e 4d orbitals of a silver ation. On the ontrary, the LUMO energyof a silver ation an be approximated by the ionization potential of a silveratom whi h is 7.5 eV, and this is well below the ondu tion band of a zeolite.Therefore, the HOMO of the omposite system is lo ated on the frameworkoxygen atoms and the LUMO onsists of the 5s orbitals on silver lusters. Ithas been suggested that the harge transfer transitions o urring between theframework HOMO and silver LUMO are responsible for the opti al propertiesof Ag/zeolite systems [23℄.The ex ited states of small Agn and Ag+n lusters in gas-phase were ex-amined by the equation-of-motion oupled lusters method with single anddouble ex itations (EOM-CCSD) and it was found that this method is verya urate when ompared to experiment [24℄. Unfortunately, it is very ostintensive and annot be reasonably used for bigger systems (silver lusters inzeolites). Also, a disadvantage of the EOM-CCSD is that its urrent imple-mentation in standard programs an only deal with losed-shell stru tures.To ir umvent these problems, performan e of several less expensive meth-ods was ompared with the EOM-CCSD to see if there was a su� ientlya urate and yet omputationally heaper method. A spe ial interest waspaid to the DFT and its time-dependent variation (TD-DFT) whi h ouldbe readily used in periodi al ulations.As the omplete zeoliti system is too large, an appropriate simpler modelwas needed for a systemati study of the ex ited states of silver lusters inzeolites. This model has to be as small as possible (be ause of the EOM-CCSD omputational ost), but bigger than just a gas-phase luster (so that30

Figure 3.1. Ag3F2 in C2v (left) and D2h (right) symmetry with theCCSD(T) bond lengths in angstroms. The Ag3 lusters are formallyin D3h and D∞h symmetry respe tively. harge transfer transitions an be observed). A water mole ule should havevery lose hemi al resemblan e to a zeolite framework (2p lone pairs onoxygen), but a negatively harged ligand in analogy with the AlO �4 tetrahedraof the zeolite framework is needed to ompensate for the positive harge ofthe silver lusters. The �uorine anion F � was hosen as it appeared to be asuitable ligand for our purposes.As was mentioned in the introdu tion, experiments suggest presen e ofa linear Agn+3 luster in zeolite Y. On the other hand, our theoreti al studypresented in Chapter 2 predi ted a trigonal luster. Therefore, we havefo used on two geometries of the Ag3F2 luster having C2v and D2h symme-tries depi ted in Figure 3.1. The bond distan es in the �gure were obtainedby geometry optimization with onstrained symmetry arried out at theCCSD(T) level. Both geometries were studied having two di�erent harges:the [Ag3F2℄+ luster represents the initial non-redu ed state of silver ations,while [Ag3F2℄ 0 is redu ed by one ele tron.3.1 MethodsThere exist several lasses of methods for modelling ex ited states. In thealready mentioned EOM-CCSD and TD-DFT, only transition energies areobtained while the ex ited state wavefun tions remain unknown.The multi on�gurational self- onsistent �eld method (MCSCF) and mul-tireferen e se ond-order perturbation theory (MRPT2) are examples of an-other lass of methods, in whi h also the ex ited wavefun tions are al ulated.As these wavefun tions need to be orthogonal to ea h other and to the ground31

state wavefun tion, the ex ited states have to be des ribed by more than onereferential Slater determinant, hen e the name multireferential methods.Finally, also the on�guration intera tion with singles (CIS) and its unre-stri ted version (UCIS) were used for modelling the ex ited states. These twomethods are quite simple and provide generally only qualitative des ription.When a symmetry is present in the system, all states of a given irredu iblerepresentation (irrep) are automati ally orthogonal to all states of all otherirreps. This enables even single-referen e methods to be used for ex itedstates by al ulating the lowest lying state of ea h irrep. Unfortunately, ex- ited singlet states and some doublet states always have to be des ribed by atleast two on�gurations (typi ally |↓↑〉 , |↑↓〉 and |↓↑↑〉 , |↑↓↓〉, respe tively).To be able to use single-referen e methods even in these ases, orrespond-ing triplet and quadruplet states an be al ulated. They have lower energy,however, qualitative information an be still obtained. Behaviour of followingsingle-referen e methods was investigated: the Møller�Plesset se ond-orderperturbation theory (MP2), the unrestri ted CCSD built upon restri tedHartree-Fo k orbitals (RHF-UCCSD) and the already mentioned DFT.Computational detailsA pseudopotential on silver atoms was employed in all al ulations. TheStuttgart RSC 1997 ECP28MWB was used together with the orresponding(8s7p6d2f1g)/[6s5p3d2f1g] basis set [25℄. This leaves 19 ele trons (4s24p64d105s1) on ea h silver atom. The standard -pVTZ basis set was used for�uorine atoms [26℄. After some preliminary tests, it was found out that f andg fun tions on silver atoms and d and f fun tions on �uorine atoms ould beomitted without deteriorating the results.The Perdew�Burke�Ernzerhof (PBE) fun tional was used in DFT al u-lations [20℄. In orrelated methods, just the 1s ele trons of �uorine werefrozen. The a tive spa es used in MCSCF/MRPT2 will be dis ussed later,sin e their onstru tion results from nontrivial ele troni stru ture of thesystem.3.2 ResultsFirst, the ele troni stru ture of the [Ag3F2℄+ luster (no 5s ele trons) hav-ing the D2h symmetry (linear silver luster) will be des ribed. The middle(between �uorine atoms) and outer silver atoms will be labeled as Agmid andAg out respe tively.The o upied valen e orbitals are separated into two blo ks. The lower32

blo k onsists of 2 F(2s) orbitals at about −42.3 eV and an be negle ted inthe study of low-energy transitions. The higher blo k onsists of 21 mole ularorbitals having energies from −21.2 eV to −15.4 eV (HOMO energy) that areformed from 15 Ag(4d) orbitals and 6 F(2p) orbitals: 3 Agmid(4d) orbitalsare highest in energy, below are 3 F(2p) orbitals followed by 2 remainingAgmid(4d) orbitals. The remaining lowest 13 orbitals of the higher blo k aremixed from 10 Ag out(4d) orbitals and 3 remaining F(2p) orbitals, and theydo not ontribute to any low lying transitions.The lowest lying blo k of uno upied orbitals onsists of the antibonding(−5.1 eV, LUMO energy) and bonding (0.4 eV above) ombinations of twoAg out(5s) orbitals. The se ond blo k of uno upied orbitals begins at−2.6 eVand onsists of one remaining Agmid(5s) orbital and the Ag(5p) orbitals onall silver atoms. This blo k of orbitals will be referred to as Ag(5p) despitethe fa t that one of them is a 5s orbital.The ele troni stru ture is very similar in ase of the C2v symmetry andthe only qualitative di�eren e is that the order of two Ag out(5s) orbitals isreversed and they are more separated�the bonding ombination is 2.0 eVbelow the antibonding.Independently on the symmetry, the ele troni stru ture is also retainedafter adding one ele tron to the LUMO, thus redu ing the system and makingit neutral.A tive spa esSlater determinants reated by ex itations within the a tive spa e are presentin the wavefun tion of the MCSCF/MRPT2. Following orbitals were in- luded in our a tive spa e (energy ordering from lowest to highest): (i) threehighest o upied F(2p) orbitals, (ii) three highest o upied Agmid(4d) or-bitals, (iii) both Ag out(5s) orbitals and (iv) three Ag(5p) orbitals. In total,the a tive spa e onsisted of 11 orbitals with 12 ele trons (in ase of hargedsystem), whi h is denoted (12,11). All possible on�gurations within it wereallowed, making it a omplete a tive spa e (CAS).To examine the role of two low lying Agmid(4d) orbitals, they were in- luded in the se ond version of an a tive spa e. This results in the (16,13)a tive spa e whi h is too large to onsider all possible ex itations within it.Therefore, they were restri ted to single and double ex itations, making ita restri ted a tive spa e (RAS). The spe i� ation of the a tive spa es is inTable 3.1. The MCSCF/MRPT2 methods used with omplete and restri teda tive spa es are usually abbreviated as CASSCF/CASPT2 and RASSCF/R-SPT2 respe tively, and this onvention is also used in this work.33

Table 3.1. Spe i� ation of a tive spa es used in CASSCF/CASPT2 and RASS-CF/RSPT2 al ulations. O upied orbitals Uno upied orbitalsSymmetry Spa e F(2p) Agmid(4d) Ag out(5s) Ag(5p)D2h CAS(12,11) 6b1u6b2u4b3u 10ag3b1g4b3g 11ag7b1u 12ag7b1u4b2gRAS(16,13) +9ag3b2gC2v CAS(12,11) 13a18b19b2 14a19b15a2 15a110b2 16a117a110b1RAS(16,13) +12a18b23.2.1 TransitionsThe obtained transition energies of the [Ag3F2℄+ and [Ag3F2℄ 0 lusters having

D2h and C2v symmetries are presented in Tables 3.2�3.5. The overall resultsfor the system having D2h symmetry (linear silver luster) are dis ussed �rst,followed by dis ussion of the di�eren es found for lower symmetry (C2v)stru tures. Then, performan e of individual methods is ommented.In the [Ag3F2℄+ luster, there are no 5s ele trons and the lowest lying tran-sitions have leading ex itations from the Agmid(4d) to uno upied Ag out(5s)orbitals. They are hara terized by energies of 3.8 eV. The most importantfa t is that no lower lying harge transfer ex itations Ag←F(2p) were found,ruling out their presen e in the visible spe trum (390�750 nm, 1.6�3.2 eV).The harge transfer ex itations were present only as ontributions to highertransitions beginning with the 18th ex ited state at about 7.6 eV (result ofless reliable CIS al ulations).There is one 5s ele tron in the [Ag3F2℄ 0 luster and it is responsible fortwo additional transition types. The Ag out(5s)←Ag out(5s) transition o ursat 0.7 eV. This orresponds to the ex itation of the 5s ele tron from thelower to higher Ag out(5s) orbital. The Ag(5p)←Agout(5s) transitions (3.7 eV) orrespond to the ex itation of the 5s ele tron to the Ag(5p) orbitals. Again,no harge transfer transitions were present up to 5.6 eV (CIS al ulations).The same qualitative pi ture as above is obtained in ase of C2v symmetry(trigonal silver luster), but the transition energies di�er signi� antly. Wewill dis uss the results for the redu ed [Ag3F2℄0. The Ag out(5s)←Agmid(4d)transitions o ur at ∼2.6 eV, i.e., ∼1.0 eV lower than in ase of the D2hsymmetry. On the ontrary, the Ag out(5s)←Ag out(5s) transition o urs at1.7 eV whi h is 1.0 eV higher. Only the Ag(5p)←Ag out(5s) transitions remainat similar energies after hanging the symmetry. The shift of energies isvery important as it makes the transitions falling into the visible range ofspe trum. Impli ations of the above results for photoa tivity of silver-dopedzeolites will be dis ussed in Se tion 3.3.34

Table 3.2. Ex ited states of [Ag3F2℄+ luster having D2h symmetry.Transition energy (eV)Transition State Multipli itya CIS EOM-CCSD CASSCF CASPT2 RASSCF RSPT2 MP2 RHF-UCCSD DFT TD-DFTAg out(5s)←Agmid(4d) 1Au S 6.34 3.67 4.01 3.81 4.79 3.85 � � � 1.32T � � � 3.82 � 3.85 4.61 3.91 2.13 �1B1u S 5.35 3.85 4.26 4.02 4.82 3.98 � � � 2.48T � � � 3.29 � 3.29 3.80 3.41 2.17 �1B2u S 6.37 3.87 4.05 4.19 4.72 4.18 � � � 1.84T � � � 4.02 � 4.03 � � 2.19 �1B1g S 5.68 4.04 4.54 4.28 5.08 4.20 � � � 1.97T � � � 3.96 � 4.02 4.66 4.03 2.34 �1B3g S 5.68 4.06 4.04 4.44 4.72 4.42 � � � 2.18T � � � 4.31 � 4.31 4.75 4.09 2.48 �2Ag S 5.53 4.44 � � � � � � � 2.79T � � � � � 3.68 4.05 3.70 2.59 �a S for singlet, T for tripletTable 3.3. Ex ited states of [Ag3F2℄ 0 luster having D2h symmetry.Transition energy (eV)Transition State Multipli itya CIS CASSCF CASPT2 RASSCF RSPT2 MP2 RHF-UCCSD DFT TD-DFTAg out(5s)←Ag out(5s) 1Ag D 0.63 0.54 0.71 0.78 0.54 0.66 0.63 0.55 0.95Ag out(5s)←Agmid(4d) D 2.87 � � � � � � � 2.241Au D 3.72 3.02 3.37 4.21 3.25 � � � 1.73Q � � 3.29 � 3.20 3.49 3.01 2.31 �1B2u D 3.79 3.34 3.40 4.41 3.13 � � � 1.89Q � � 3.32 � 3.41 3.49 3.00 2.32 �Ag(5p)←Ag out(5s) 1B3u D 2.84 � � � � � � � �1B2g D 3.37 3.16 3.79 3.36 3.68 3.66 3.64 4.17 �3B1u D 3.45 � � � � � � � �2B2u D 3.53 � � � � � � � �2Ag D 3.65 � � � � � � � 3.471B3g D 3.78 � � � � � � � �a D for singlet, Q for triplet 35

Table 3.4. Ex ited states of [Ag3F2℄+ luster having C2v symmetry.Transition energy (eV)Transition State Multipli itya CIS EOM-CCSD CASSCF CASPT2 RASSCF RSPT2 MP2 RHF-UCCSD DFT TD-DFTAg out(5s)←Agmid(4d) 1A2 S 3.95 2.40 2.44 2.46 3.61 2.54 � � � 1.10T � � � 2.35 � 2.46 2.95 2.23 1.38 �1B1 S 4.01 2.48 2.59 2.55 3.72 2.64 � � � 1.24T � � � 2.34 � 2.41 2.92 2.19 1.40 �2A1 S 4.03 3.23 � � � � � � � 1.96T � � � 2.00 � 1.86 2.26 1.86 1.66 �1B2 S 5.82 3.96 3.59 3.53 5.64 4.21 � � � 1.99T � � � 3.50 � 4.08 3.64 3.67 2.27 �a S for singlet, T for triplet

Table 3.5. Ex ited states of [Ag3F2℄ 0 luster having C2v symmetry.Transition energy (eV)Transition State Multipli itya CIS CASSCF CASPT2 RASSCF RSPT2 MP2 RHF-UCCSD DFT TD-DFTAg out(5s)←Ag out(5s) 1B2 D 2.35 2.95 1.64 2.61 1.72 1.84 1.83 1.34 1.39Ag out(5s)←Agmid(4d) 2A1 D 3.21 � � � � � � � 2.011A2 D 4.41 3.62 2.74 5.04 2.59 2.99 2.60 1.65 1.16Ag(5p)←Ag out(5s) 1B1 D 4.01 4.79 3.57 4.50 3.57 3.88 3.86 3.98 4.043A1 D 4.05 � � � � � � � 3.944A1 D 4.42 � � � � � � � �a D for doublet, Q for quadruplet

36

Performan e of methodsIt an be seen that MRPT2 results di�er by no more than 0.3 eV from theEOM-CCSD and provide quantitatively orre t des ription of the ex itedstates. While the CASSCF and CASPT2 give similar results, the RASSCFgives a poor des ription that is improved signi� antly by perturbation theory(RSPT2). It is lear from this that the onstraints imposed on ex itations inthe RAS are too strong. A very good agreement between the CASPT2 andRSPT2 shows that two lower lying Agmid(4d) orbitals (whi h are present inthe RAS but not in the CAS) are not important for des ription of the ex itedstates.As for the single-referen e methods, the MP2 systemati ally overestimatesthe Ag out(5s)←Agmid(4d) transitions, while it gives very good results for twoother types of the transitions. The RHF-UCCSD (whi h has the same levelof a ura y as the EOM-CCSD) is in good agreement with the CASPT2 inalmost all ases.DFT methods perform very poorly. The single-referen e DFT gives atleast qualitatively good predi tions: order of the ex ited states is orre tand transitions of the same type have similar energies. But behaviour of theTD-DFT is ompletely errati (many states were not even found). The dif-feren e ECASPT2 −EDFT depends on the type of a transition: it is 0.1�0.3 eVfor Ag out(5s)←Ag out(5s) transitions, lower than −1.0 eV for Ag out(5s)←Agmid(4d) transitions and greater than 0.5 eV for Ag(5p)←Agout(5s) transi-tions.3.3 Dis ussionThe al ulated transitions with their energies and orresponding wavelengthsare summarized in Table 3.6. No harge transfer transitions of type Ag←Ffrom the ligand to silver orbitals were found. This disagrees with the resultsof Calzaferri et. al., who arried out �mole ular orbital al ulations� of non-spe i�ed hara ter with a ontradi ting out ome [23℄.No transitions in the visible range were found in ase of the linear silver luster. The Ag(5s)←Ag(5s) transition is lo ated in the infrared spe trum,while two other types of transitions have UV wavelengths. On the on-trary, the redu ed trigonal luster posses two potentially visible transitions:Ag(5s)←Ag(5s) at 730 nm and Ag(5s)←Ag(4d) at 460 nm. Cremer et. al.identi�ed two �uores ent bands in Ag-doped zeolites: the �rst present onlyin zeolite A at about 690 nm and the se ond in zeolites A, X and Y at about550 nm. The latter transitions were assigned to the Agn+3 lusters. Results37

Table 3.6. CASPT2 ex itation energies in eV (light wavelengths in nm)depending on the type of transition, the symmetry and the number of 5sele trons. TransitionsSymmetry 5s a Ag out(5s)←Ag out(5s) Ag out(5s)←Agmid(4d) Ag(5p)←Ag out(5s)D2h 0 � 3.8 (330) �1 0.7 (1800) 3.4 (370) 3.7 (340)C2v 0 � 2.4 (520) �1 1.7 (730) 2.7 (460) 3.6 (350)a Number of 5s ele trons.presented above suggest that trigonal silver lusters in a SOD ould be re-sponsible for experimentally observed transitions, however, it must be notedthat the al ulations of the ex ited states are still only preliminary.To on�rm or disprove the ideas, a further study is needed. First of all,[Ag3℄n+ lusters with two and three 5s ele trons should be investigated, to over the whole range from silver ations to ompletely redu ed silver lusters.E�e t of identity (and even presen e) of the ligands on the silver transitionsshould also be studied. If there is no or very small dependen e, the gas-phase luster is a tually a very good model for silver lusters in zeolites, sin e thereare probably no harge transfer transitions from the zeolite framework.It was shown that the MRPT2 (mu h heaper than the EOM-CCSD)is fully apable of des ription of the ex ited states of silver lusters. Thisenables a potential use of mu h bigger models for the zeolite framework:a double-6-ring or a part of the sodalite age. On the ontrary, the PBEfun tional has very poor performan e. A thorough study of its behaviourwould be needed to see if there is a han e of some systemati orre tions.Also, another DFT fun tionals ould be tried.In summary, our results suggests that ele troni transitions in visiblerange are more likely to be observed for the trigonal silver lusters than forthe linear lusters. This on lusion ontradi ts some of the experimental workthat proposes the formation of linear lusters. While our results obtained forthe ex ited states in Ag3 lusters ontradi t these experimental on lusions,they are onsistent with the results on stability of the ground-state silver lusters in zeolite Y presented in the previous hapter.

38

Con lusionsThe aim of this ba helor's thesis was to investigate silver lusters in zeolites.It was motivated by experimental results suggesting that silver-ex hangedzeolites ould have great appli ations in light te hnology. To a hieve thegoal, the study has been divided into three parts.First, we have developed and implemented an algorithm for geometryoptimizations. Our algorithm proved to be mu h more e� ient for our taskthan the algorithms available in standard ommer ial programs. Overall,the omputational time needed for geometry optimizations was signi� antlylowered.Armed with the algorithm, we have sear hed for stable Agn+3 lusters inzeolite Y. The original idea about linear lusters in a double-6-ring omingfrom the literature was not proved. Our results suggest that the linear lusteris not formed. Instead, non-redu ed silver ations do not intera t at all, andafter redu tion a trigonal luster in the sodalite age is formed. All this isvery important sin e the shape of the luster has a dominant e�e t on itsopti al properties; this was examined in the last part of the thesis.Finally, we have al ulated the opti al properties of silver lusters in asimple model where the zeolite framework was represented by two �uorineanions. The idea of harge transfer transitions from ligands to silver atomswas not proved; all the al ulated transitions take pla e on the silver lus-ter. Also, it was shown that the spe tra depend signi� antly on a shape ofthe luster. While no transitions in the visible spe trum o ur in the linear luster, the redu ed trigonal luster has two types of potentially visible tran-sitions: Ag(5s)←Ag(4d) and Ag(5s)←Ag(5s). This is in agreement with theexperimental results, though it annot be ruled out as a mere oin iden eat this stage of study. It was also found out that the DFT methods give avery poor des ription of ex ited states of silver lusters. If they are to beused nevertheless, a further study is needed to sear h for some systemati orre tions.This thesis presents a �rst systemati study of silver lusters in zeolites.It gave us an initial theoreti al insight into the topi , it suggested importan e39

of the redu ed trigonal silver in photoa tivity of Ag/zeolite systems, and itraised new questions that will be investigated in the future.

40

Bibliography[1℄ Sun, T.; Se�, K. Silver Clusters and Chemistry in Zeolites. Chem. Rev.1994, 94, 857�870.[2℄ De Cremer, G.; Antoku, Y.; Roe�aers, M. B. J.; Sliwa, M.; Van Noyen,J.; Smout, S.; Hofkens, J.; De Vos, D. E.; Sels, B. F.; Vos h, T. Photoa -tivation of Silver-Ex hanged Zeolite A. Angew. Chem. Int. Ed. 2008, 47,2813�2816.[3℄ De Cremer, G.; Coutiño-Gonzales, E.; Roe�aers, M. B. J.; Moens, B.;Ollevier, J.; Van der Auweraer, M.; S hoonhydt, R.; Ja obs, P. A.;De S hryver, F. C.; Hofkens, J.; De Vos, D. E.; Sels, B. F.; Vos h,T. Chara terization of Fluores en e in Heat-Treated Silver-Ex hangedZeolites. J. Am. Chem. So . 2009, 131, 3049�3056.[4℄ De Cremer, G.; Coutiño-Gonzales, E.; Roe�aers, M. B. J.; De Vos, D. E.;Hofkens, J.; Vos h, T.; Sels, B. F. In Situ Observation of the EmissionChara teristi s of Zeolite-Hosted Silver Spe ies During Heat Treatment.ChemPhysChem 2010, 11, 1627�1631.[5℄ De Cremer, G.; Sels, B. F.; Hotta, J.; Roe�aers, M. B. J.;Bartholomeeusen, E.; Coutiño-Gonzales, E.; Valt hev, V.; De Vos, D.E.; Vos h, T.; Hofkens, J. Opti al En oding of Silver Zeolite Mi ro ar-riers. Adv. Mater. 2010, 22, 957�960.[6℄ Hrat hian, H. P.; S hlegel, H. B. Finding minima, transition states,and following rea tion pathways on ab initio potential energy surfa es.In Theory and appli ations of omputational hemistry: the �rst fortyyears; Dykstra, C. et al., Ed.; Elsevier: Amsterdam, 2005; p. 195.[7℄ S hlegel, H. B. Optimization of Equilibrium Geometries and TransitionStru tures. J. Comp. Chem. 1982, 3, 214�218[8℄ Banerjee, A.; Adams, N.; Simons, J.; Shepard, R. Sear h for StationaryPoints on Surfa es. J. Phys. Chem. 1985, 89, 52�57.[9℄ Flet her, R. Pra ti al Methods of Optimization, 2nd ed.; Wiley: NewYork, 1987. 41

[10℄ Bo�ll, J. M. An updated Hessian formula for optimizing transition stru -tures whi h expli itly ontains the potential stru ture of the desiredtransition ve tor. Chem. Phys. Lett. 1996, 260, 359�364.[11℄ Peng, C.; Ayala, P. Y.; S hlegel, H. B.; Frish, M. J. Using Redundant In-ternal Coordinates to Optimize Equilibrium Geometries and TransitionStates. J. Comput. Chem 1996, 17, 49�56.[12℄ Swart, M.; Bi kelhaupt, F. M. Optimization of Strong and Weak Coor-dinates. Int. J. Quantum Chem. 2006, 106, 2536�2544.[13℄ Eaton, J. W. GNU O tave Manual; Network Limited Ltd, 2002.[14℄ Olson, D. H. The rystal stru ture of dehydrated NaX. Zeolites 1995,15, 439�443.[15℄ Frising, T.; Le�aive, P. Extraframework ation distributions in X andY faujasite zeolites: A review. Mi ropor. Ma ropor. Mat. 2008, 114,27�63.[16℄ Gellens, L. R.; Mortier, W. J.; S hoonheydt, R. A.; Uytterhoeven, J.B. The Nature of the Charged Silver Clusters In Dehydrated Zeolites ofType A. J. Phys. Chem. 1981, 85, 2783�2788.[17℄ Sierka, M.; Sauer, J. Proton Mobility in Chabazite, Faujasite, and ZSM-5 Zeolite Catalysts. Comparison Based on ab Initio Cal ulations. J.Phys. Chem. 2001, 105, 1603�1613.[18℄ Kresse, G.; Hafner, J. Ab initio mole ular dynami s for liquid metals.Phys. Rev. B 1995, 47, 558�561.[19℄ Kresse, G.; Furthmüller, J. E� ient iterative s hemes for ab initio total-energy al ulations using a plane-wave basis set. Phys. Rev. B 1996, 54,11169�11186.[20℄ Perdew, J. P.; Burke, K.; Ernzerhof, M. Generalized Gradient Approxi-mation Made Simple. Phys. Rev. Lett. 1996, 77, 3865�3868.[21℄ Blö hl, P. E. Proje tor augmented-wave method. Phys. Rev. B 1994,50, 17953�17979.[22℄ Kresse, G.; Joubert, D. From ultrasoft pseudopotentials to the proje toraugmented-wave method. Phys. Rev. B 1999, 59, 1758�1775.[23℄ Calzaferri, G.; Leiggener, C.; Glaus, S.; S hür h, D.; Kuge, K. Theele troni stru ture o Cu+, Ag+, and Au+ zeolites. Chem. So . Rev.2003, 32, 29�37.[24℄ Bona£i¢-Koute ký, V.; Pittner, J.; Boiron, M.; Fantu i, P. An a uraterelativisti e�e tive ore potential for ex ited states of Ag atom: Anappli ation for studying the absorption spe tra of Agn an Ag+n lusters.J. Chem. Phys. 1999, 110, 3876�3886.42

[25℄ Andrae, D.; Häuÿermann, U.; Dolg, M.; Stoll, H.; Preuÿ, H. Energy-adjusted ab initio pseudopotentials for the se ond and third row transi-tion elements. Theor. Chim. A ta 1990, 77, 123�141.[26℄ Dunning, Jr., T. H. Gaussian basis sets for use in orrelated mole ular al ulations. I. The atoms boron through neon and hydrogen. J. Chem.Phys 1898, 90, 1007�1023.

43

Appendix AProgram listingThe listing of all fun tion �les needed to run the geometry optimizationalgorithm in GNUO tave is presented in this appendix.Listing 1. berny.m% performs one step of Berny algorithmfun tion berny(�d)5 if exist('workspa e.mat','�le')load workspa e.matsteps = steps + 1;initial = false;else10 steps = 1;[x,atoms℄ = readX('xyz'); % initial artesian geometry and atom labelsm = size(x,1); % number of atoms[ind,rho℄ = genind(x,atoms); % generates internal oords de�nitionn = size(ind,1); % number of oords15 q = internals(x,ind);[w,types℄ = weights(x,q,ind,rho);C = zeros(n); % onstraint matrix (zeros − no onstraints)fprintf(�d,['0 Coordinates information:\n0 * Number of atoms: %g\n'...'0 * Number of internal oordinates: %g\n'...20 '0 * Number of strong bonds: %g\n' '0 * Number of weak bonds: %g\n'...'0 * Number of superweak bonds: %g\n'℄,m,n,types(1),types(2),types(3));if exist('H','�le')load Hfprintf(�d,'0 Initial Hessian loaded from �le\n');25 elseH = hessian(rho,ind);fprintf(�d,'0 Initial Hessian approximated by Swart''06 model\n');endtrust = 0.3; % initial trust radius30 initial = true;end�ush(�d); 44

B = w*Bmat(x,ind); % internal oords and Wilson matrix35 G = B*B';Gi = ginv(G); % generalized inverseif (initial && exist('strains','�le'))load strains % loads oords to be onstrained40 ns = size(strains,1); % number of onstrained oordsfor i=1:nsstr = strains(i,:);k = strmat h(str,ind);if isempty(k)45 str = str(str>0);str = [�iplr(str) zeros(1,4−length(str))℄;k = strmat h(str,ind);endC(k,k) = 1;50 endfprintf(�d,'0 * Number of onstrained oordinates: %g\n',sum(diag(C))); lear strains ns i str k qtest xdummyend55 pre = [num2str(steps) ' '℄;load e % energy of urrent stepload g % artesian gradient of urrent stepg = reshape(g',3*m,1);g = Gi*B*g; % internal gradient60 fprintf(�d,[pre 'Energy: %4.8f\n'℄,e);�ush(�d);if ~initial% hessian update65 dg = g−g1;BFGS = dg*dg'/(dq'*dg)−H*(dq*dq')*H/(dq'*H*dq); % BFGS updateMS = (dg−H*dq)*(dg−H*dq)'/(dq'*(dg−H*dq)); % MS updateQ = abs(dq'*(dg−H*dq))/(norm(dq)*norm(dg−H*dq)); % Bo�ll's oe�dH = (1−Q)*BFGS+Q*MS;70 H = H + dH;fprintf(�d,[pre 'Hessian update information:\n' ...pre '* Bo�ll''s oe� ient: %g\n' ...pre '* Maximum absolute hange: %g\n' ...pre '* RMS hange: %g\n'℄, ...75 Q,norm(diag(dH),inf),rms(dH)); lear dg BFGS MS Q dHend�ush(�d);80 if ~initial% trust region updatede = e−e1; % a tual energy hanger = de/(delP+deqP); % �et her's parameterif de == 085 r = 1; % round−o�s 45

endfprintf(�d,[pre 'Flet her''s parameter: %g\n'℄,r);if r < 0.25trust = norm(dqq)/4;90 elseif (r > .75 && onsphere)trust = 2*trust;end lear de rend95 �ush(�d);if initialgi = g; % interpolated gradient for quadrati step is a tual gradientdql = 0; % linear step is zero100 delP = 0; % predi ted energy hange from linear step is zerofprintf(�d,[pre 'No linear step was taken\n'℄);else% linear sear hf1 = g1'*dq; % gradient proje tions105 f = g'*dq;fprintf(�d,[pre 'Linear interpolation:\n'...pre '* Energies: %4.6f, %4.6f\n' pre '* Derivatives: %g, %g\n'℄,e1,e,f1,f);[t ei℄ = �tquarti (e,e1,−f,−f1); % quarti �tif isnan(t)110 [t ei℄ = �t ubi (e,e1,−f,−f1); % ubi �tif (isnan(t) || ~(t > 0 && t < 1) || e == e1)if e <= e1% no linear step is takenfprintf(�d,[pre '* No linear step was taken\n'℄);115 t = 0;ei = e;else% linear step ba k to best geometry is takenfprintf(�d,[pre '* Wrong quadrati step, going ba k\n'℄);120 t = 1;ei = e1;endelsefprintf(�d,[pre '* Cubi interpolation was performed: t=%g\n'℄,−t);125 endelsefprintf(�d,[pre '* Quarti interpolation was performed: t=%g\n'℄,−t);enddql = −t*dq; % linear step130 delP = ei−e; % predi ted energy hange from linear stepif (t~=0 && t~=1)fprintf(�d,[pre 'Predi ted energy hange from linear step: %g\n'℄,delP);endgi = (1−t)*g+t*g1; % interpolation of gradient135 %if (t~=0 && t~=1)% gi = gi − ((dql'*gi)/norm(dql)^2)*dql; % subtra t tangent omponent%end lear f1 f t ei 46

end140 �ush(�d);% quadrati step using RFO and trust regionproj = G*Gi; % proje tor on nonredundant subspa e%proj = proj − proj*C*ginv(C*proj*C)*C*proj; % onstrained proje tor145 gp = proj*gi; % proje t gradientHp = proj*H*proj + 1e4*(eye(n)−proj); % proje t Hessianrfo = [Hp gp; gp' 0℄;% RFO matrix from urrent Hessian and interpolated gradient[eve ,ev℄ = eig(rfo); % solve eigenproblem150 [ev,P℄ = sort(real(diag(ev))); % sort eigenvaluesif ~initialdqqlast = dqq; % remember for angle al ulationenddqq = eve (:,P(1))/eve (end,P(1)); % lowest eigenve tor is regular RFO step155 dqq(end) = [℄; % remove last row (whi h is equal to 1)fprintf(�d,[pre 'Trust radius: %g\n'℄,trust);if rms(dqq) > trust % trust region riteriumsteplength = �(l) (norm((l*eye(n)−Hp)\gp)−trust);% steplength as a fun tion of lambda160 l = fzero(steplength,[−1000 ev(1)℄); % minimization on spherefprintf(�d,[pre 'Minimization on sphere: lambda=%g\n'℄,l);dqq = (l*eye(n)−Hp)\gp; % step on spheredeqP = (gp'*dqq+0.5*dqq'*Hp*dqq)/(1+dqq'*dqq); % predi ted energy hangeonsphere = true; % minimization on sphere was performed165 elsedeqP = ev(1)/2; % predi eted energy hangeonsphere = false; % regular RFO step was takenendif ~initial170 angle = 180/pi*a os(dqqlast'*dqq/norm(dqqlast)/norm(dqq));endfprintf(�d,[pre 'RMS of quadrati step: %g\n'...pre 'Predi ted energy hange from quadrati step: %g\n'℄,...rms(dqq),deqP);175 if ~initialfprintf(�d,[pre 'Angle di�eren e to last step: %g\n'℄,angle);end lear proj Hp rfo eve ev P steplength l dqqlast angle�ush(�d);180 q1 = q; % previous al ulated pointe1 = e;g1 = g;dq = dql+dqq; % total step185 % onverge e riteriathreshold = [0.45e−3 0.3e−3 1.8e−3 1.2e−3℄;values = [max(abs(gp(~diag(C)))) rms(gp(~diag(C))) max(abs(dq)) rms(dq)℄;names = {'Gradient maximum' 'Gradient RMS' 'Step maximum' 'Step RMS'};190 �ags = zeros(1,4);fprintf(�d,[pre 'Convergen e riteria:\n'℄); 47

for i=1:4if values(i) < threshold(i)b = 'OK';195 �ags(i) = 1;elseb = 'no';endfprintf(�d,[pre '* ' har(names(i)) ' is: %g, threshold: %g, ' b '\n'℄, ...200 values(i),threshold(i));endif all(�ags)fprintf(�d,[pre 'All riteria mat hed\n'℄,steps);end205 lear e g i b gp�ush(�d);[x,q℄ = red2 ar(dq,q,ind,B,Gi,w,m,x,C,�d,pre);�ush(�d);210 writeX(x,atoms,'xyz') % save artesians for energy al ulationsave workspa e.matend215 fun tion [x,q℄ = red2 ar(dq,q,ind,B,Gi,w,m,x,C,�d,pre)i = 0;q1 = q;while true % break in the loop220 dx = 0.52917720859*reshape(B'*Gi*w*dq,3,m)'; % iteration stepx = x+dx; % update artesiansqnew = internals(x,ind); % new internal oordsdq = orre t(dq−(qnew−q)); % new step in internalsq = qnew; % prepare for next iteration step225 i = i+1;if rms(dx)<1e−6, break, endif i>100fprintf(�d,[pre 'Cannot transform redundants to artesians!\n'℄);error('error')230 exitendendfprintf(�d,[pre 'Iteration #%g: rms(dx)=%g, rms(dq)=%g\n'℄,...i,rms(dx),rms(dq));235 if sum(diag(C))dq = − orre t(q−q1);i = 0;while true % break in the loopdx = 0.52917720859*reshape(B'*Gi*w*C*dq,3,m)'; % iteration step240 x = x+dx; % update artesiansqnew = internals(x,ind); % new internal oordsdq = orre t(dq−(qnew−q)); % new step in internalsq = qnew; % prepare for next iteration stepi = i+1; 48

245 if rms(dx)<1e−6, break, endif i>100fprintf(�d,[pre 'Cannot transform redundants to artesians!\n'℄);error('error')exit250 endendfprintf(�d,[pre 'Iteration #%g: rms(dx)=%g, rms(dq)=%g\n'℄,...i,rms(dx),rms(dq));end255 end Listing 2. genind.m% generates system of internal oordinates based on standard ovalent radiifun tion [ind,rho℄ = genind(x,atoms)n = length(atoms); % number of atoms5 R = dist(x); % matrix of interatomi distan es% onstru ts matrix of ovalent distan es ov = zeros(n);for i=1:n10 for j=(i+1):n ov(i,j) = radius(atoms{i})+radius(atoms{j});endend15 rho = exp(−R./ ov+1); % matrix for oordinate boostingrho = triu(rho,1)+triu(rho,1)';bond = double(R<1.3* ov); % bond matrixbond = bond+bond';20 C = onn(bond); % matrix of ovalent onne tivity% onstru ts fragmentsv = zeros(n,1);nf = 0;25 for i=1:nif v(i) == 0nf = nf+1;w = zeros(n,1);w(i) = 1;30 frag = �nd(C*w);v(frag) = 1;fragments{nf} = frag;endend35 % adds interfragment bondsfor i=1:nffor j=i+1:nfa = fragments{i}; 49

40 b = fragments{j};shdis = min(min(R(a,b)));sele t = zeros(n);sele t(a,b) = 1;Rintfrag = R.*sele t;45 Rintfrag(Rintfrag == 0) = 1000;bondadd = double( Rintfrag<1.2*shdis | Rintfrag<2 );bond = bond+bondadd+bondadd';endend50 % generates the list of oordinates oords = [℄;for i=1:nstep1 = bond(i,:);55 bonds = �nd(step1(i+1:n))+i;n = length(bonds);if n > 0 oords = [ oords; [ones(n ,1)*i bonds' zeros(n ,2)℄℄;end60 for j=1:nif ~step1(j), ontinue, endstep2 = bond(j,:);step2(i) = 0;angles = �nd(step2(i+1:n))+i;65 n = length(angles);if n > 0 oords = [ oords;...[ones(n ,1)*i ones(n ,1)*j angles' zeros(n ,1)℄℄;end70 for k=1:nif ~step2(k), ontinue, endif angle(x,i,j,k)>pi−1e−6, ontinue, endstep3 = bond(k,:);step3([i j℄) = 0;75 dihedrals = �nd(step3);if ~isempty(dihedrals)[ oords,dihedrals℄ = ladder( oords,n,step3,bond,dihedrals,x,i,j,j,k);dihedrals = dihedrals(dihedrals>i);n = length(dihedrals);80 if n > 0 oords = [ oords;...[ones(n ,1)*i ones(n ,1)*j ones(n ,1)*k dihedrals'℄℄;endend85 endendendind = oords;90 end% re ursive fun tion; dis ards linear dihedrals and goes one step up the linear ladder50

fun tion [ oords,dihedrals℄ = ladder( oords,n,step,bond,dihedrals,x,i,j,k,l)for m=1:n95 if ~step(m), ontinue, endif angle(x,k,l,m)>pi−1e−6dihedrals(dihedrals==m) = [℄;step2 = bond(m,:);step2([i l℄) = 0;100 dihedrals2 = �nd(step2);if ~isempty(dihedrals2)[ oords,dihedrals2℄ = ladder( oords,n,step2,bond,dihedrals2,x,i,j,l,m);dihedrals2 = dihedrals2(dihedrals2>i);n = length(dihedrals2);105 if n > 0 oords = [ oords;...[ones(n ,1)*i ones(n ,1)*j ones(n ,1)*m dihedrals2'℄℄;endend110 endendend Listing 3. internals.m% al ulates internal oordinates, see Bmat.m3 fun tion q = internals(x, ind)m = size(x, 1);n = size(ind, 1);q = zeros(n, 1);x = x/0.52917720859;8 for i=1:nif ind(i,3) == 0R = r(x(ind(i,1),:), x(ind(i,2),:));q(i) = R;else13 if ind(i,4) == 0Phi = ang(x(ind(i,1),:), x(ind(i,2),:), x(ind(i,3),:));elsePhi = dih(x(ind(i,1),:), x(ind(i,2),:), ...x(ind(i,3),:), x(ind(i,4),:));18 endq(i) = Phi;endendend23 fun tion R = r(r1, r2)v = r1−r2;R = norm(v);end28 fun tion Phi = ang(r1, r2, r3)v1 = r1−r2; 51

v2 = r3−r2;Phi = a os(v1*v2'/norm(v1)/norm(v2));33 endfun tion Phi = dih(r1, r2, r3, r4)v1 = r1−r2;v2 = r4−r3;38 w = r3−r2;ew = w/norm(w);a1 = v1−(v1*ew')*ew;a2 = v2−(v2*ew')*ew;sgn = sign(det([v2; v1; w℄));43 if ~sgnsgn = 1;ends = a1*a2'/norm(a1)/norm(a2);if s <= −148 s = −1;elseif s >= 1s = 1;endPhi = a os(s )*sgn;53 end Listing 4. weights.m1 fun tion [w,bonds℄ = weights(x,q,ind,rho)w = zeros(size(q,1));n = size(q,1);f = 0.12;strong = 0;6 weak = 0;superweak = 0;for i=1:nif ind(i,3)==0w(i,i) = rho(ind(i,1),ind(i,2));11 if w(i,i) > 0.7strong = strong+1;elseif w(i,i) > 0.3weak = weak+1;else16 superweak = superweak+1;endelseif ind(i,4)==0w(i,i) = sqrt(rho(ind(i,1),ind(i,2))*rho(ind(i,2),ind(i,3)))...*(f+(1−f)*sin(q(i)));21 elseth1 = angle(x,ind(i,1),ind(i,2),ind(i,3));th2 = angle(x,ind(i,2),ind(i,3),ind(i,4));w(i,i) = (rho(ind(i,1),ind(i,2))*rho(ind(i,2),ind(i,3))...*rho(ind(i,3),ind(i,4)))^(1/3)...26 *(f+(1−f)*sin(th1))*(f+(1−f)*sin(th2));end 52

endbonds = [strong weak superweak℄;end Listing 5. hessian.mfun tion H = hessian(rho,ind)n = size(ind,1);H = zeros(n);for i=1:n5 if ind(i,3)==0H(i,i) = 0.45*rho(ind(i,1),ind(i,2));elseif ind(i,4)==0H(i,i) = 0.15*rho(ind(i,1),ind(i,2))*rho(ind(i,2),ind(i,3));else10 H(i,i) = 0.005*rho(ind(i,1),ind(i,2))*rho(ind(i,2),ind(i,3))...*rho(ind(i,3),ind(i,4));endendend Listing 6. Bmat.m1 % al ulates Wilson matrixfun tion B = Bmat(x, ind)m = size(x, 1); % number of atomsn = size(ind, 1); % number of oords6 B = zeros(n, 3*m);x = x/0.52917720859; % angstroms to bohrsfor i=1:nif ind(i,3) == 0grad = r(x(ind(i,1),:), x(ind(i,2),:));11 k = 2;elseif ind(i,4) == 0grad = ang(x(ind(i,1),:), x(ind(i,2),:), x(ind(i,3),:));k = 3;else16 grad = dih(x(ind(i,1),:), x(ind(i,2),:), ...x(ind(i,3),:), x(ind(i,4),:));k = 4;endfor j=1:k21 pos = 1+3*(ind(i,j)−1);B(i,pos:(pos+2)) = grad(j,:);endendend26 fun tion grad = r(r1, r2)v = r1−r2;grad = zeros(2,3);grad(1,:) = v/norm(v); 53

31 grad(2,:) = −v/norm(v);endfun tion grad = ang(r1, r2, r3)v1 = r1−r2;36 v2 = r3−r2;Phi = a os(v1*v2'/norm(v1)/norm(v2));grad = zeros(3,3);if norm(Phi)>pi−1e−6 % limit ase for phi=180grad(1,:) = (pi−Phi)/(2*norm(v1)^2)*v1;41 grad(3,:) = (pi−Phi)/(2*norm(v2)^2)*v2;grad(2,:) = (1/norm(v1)−1/norm(v2))*(pi−Phi)/(2*norm(v1))*v1;elsegrad(1,:) = ot(Phi)*v1/norm(v1)^2 − v2/(norm(v1)*norm(v2)*sin(Phi));grad(3,:) = ot(Phi)*v2/norm(v2)^2 − v1/(norm(v1)*norm(v2)*sin(Phi));46 grad(2,:) = (v1+v2)/(norm(v1)*norm(v2)*sin(Phi)) ...− ot(Phi)*(v1/norm(v1)^2+v2/norm(v2)^2);endend51 fun tion grad = dih(r1, r2, r3, r4)v1 = r1−r2;v2 = r4−r3;w = r3−r2;ew = w/norm(w);56 a1 = v1−(v1*ew')*ew;a2 = v2−(v2*ew')*ew;sgn = sign(det([v2; v1; w℄)); % sign of the dihedral angleif ~sgn % for planar systems,sgn = 1; % do nothing61 ends = a1*a2'/norm(a1)/norm(a2);if s <= −1 % orre tion of ma hine pre isions = −1;elseif s >= 166 s = 1;endPhi = a os(s )*sgn;grad = zeros(4,3);if norm(Phi)>pi−1e−6 % limit ase for phi=18071 g = ross(w,a1);g = g/norm(g);grad(1,:) = g/norm(g)/norm(a1);grad(4,:) = g/norm(g)/norm(a2);A = v1*ew'/norm(w);76 B = v2*ew'/norm(w);grad(3,:) = −((1+B)/norm(a2)+A/norm(a1))*g;grad(2,:) = −((1−A)/norm(a1)−B/norm(a2))*g;elseif norm(Phi)<1e−6 % limit ase for phi=0g = ross(w,a1);81 g = g/norm(g);grad(1,:) = g/norm(g)/norm(a1);grad(4,:) = −g/norm(g)/norm(a2); 54

A = v1*ew'/norm(w);B = v2*ew'/norm(w);86 grad(3,:) = ((1+B)/norm(a2)−A/norm(a1))*g;grad(2,:) = −((1−A)/norm(a1)+B/norm(a2))*g;elsegrad(1,:) = ot(Phi)*a1/norm(a1)^2 − a2/(norm(a1)*norm(a2)*sin(Phi));grad(4,:) = ot(Phi)*a2/norm(a2)^2 − a1/(norm(a1)*norm(a2)*sin(Phi));91 A = v1*ew'/norm(w);B = v2*ew'/norm(w);grad(3,:) = ((1+B)*a1+A*a2)/(norm(a1)*norm(a2)*sin(Phi)) ...− ot(Phi)*((1+B)*a2/norm(a2)^2+A*a1/norm(a1)^2);grad(2,:) = ((1−A)*a2−B*a1)/(norm(a1)*norm(a2)*sin(Phi)) ...96 − ot(Phi)*((1−A)*a1/norm(a1)^2−B*a2/norm(a2)^2);endend Listing 7. �tquarti .m% �ts onstrained quarti polynomial to fun tion values and derivatives at x=0,12 % quarti polynomial is onstrained su h that it's 2nd derivative is zero at just one% point. this ensures that it has just one maximum or one minimum. no su h or two% su h quarti polynomials always exist. from the two, the one with lower minimum is% hosen% returns position and fun tion value of minimum or NaN if �t fails or has a maximum7 fun tion [x y℄ = �tquarti (y0, y1, g0, g1)D = −(g0+g1)^2 − 2*g0*g1 + 6*(y1−y0)*(g0+g1) − 6*(y1−y0)^2;% dis riminant of d^2y/dx^2=0if D < 1e−1112 x = NaN;y = NaN;returnelsem = −5*g0 − g1 − 6*y0 + 6*y1;17 p1 = helper(y0, y1, g0, g1, .5*(m+sqrt(2*D)));p2 = helper(y0, y1, g0, g1, .5*(m−sqrt(2*D)));if (p1(1) < 0 && p2(1) < 0)x = NaN;y = NaN;22 returnend[x1 min1℄ = min(p1);[x2 min2℄ = min(p2);if min1 < min2 % hoose the one with lower minimum27 x = x1;y = min1;elsex = x2;y = min2;32 endendend 55

fun tion p = helper(y0, y1, g0, g1, )37 a = + 3*(y0−y1) + 2*g0 + g1;b = −2* − 4*(y0−y1) − 3*g0 − g1;p = [a b g0 y0℄;end42 % returns minimum of onstrained quarti fun tion [x y℄ = min(p)r = roots(polyder(p));reals = ( onj(r).*r−r.*r == 0);if sum(reals) == 147 x = r(reals);elsex = r(logi al((r==max(−abs(r)))+(r==−max(−abs(r)))));endy = polyval(p, x);52 end Listing 8. �t ubi .m% �ts ubi polynomial to fun tion values and derivatives at x=0,12 % returns position and fun tion value of minimum or NaN if polynomial% doesn't have extremafun tion [x y℄ = �t ubi (y0,y1,g0,g1)a = 2*(y0−y1)+g0+g1;7 b = −3*(y0−y1)−2*g0−g1;p = [a b g0 y0℄; % �tted polynomialr = roots(polyder(p)); % stationary pointsif isreal(r) % has extrema?r = sort(r);12 if p(1)>0x = r(2);max = r(1);elsex = r(1);17 max = r(2);endy = polyval(p,x);if ( (max > 0 && max < 1) || abs(x−.5)>abs(max−.5) )x = NaN;22 y = NaN;endelsex = NaN;y = NaN;27 endend Listing 9. ginv.m% generalized inverse of matrix2 56

fun tion A = ginv(A)[V,D℄ = eig(A);D(abs(D)>1e−10) = 1./D(abs(D)>1e−10);A = real(V*D*V');7 end Listing 10. onn.mfun tion A = onn(A)2 n = length(A);while trueB = logi al(A*A.*~eye(n)+A)+0;if A == Bbreak7 elseA = B;endendA = A+eye(n);12 end Listing 11. radius.m% returns standard ovalent radius of atom3 fun tion x = radius(s)swit h s ase 'H'x = 0.31; ase 'B'8 x = 0.84; ase 'C'x = 0.73; ase 'N'x = 0.71;13 ase 'O'x = 0.66; ase 'F'x = 0.57; ase 'Cl'18 x = 1.02; ase 'Ag'x = 1.45; ase 'Si'x = 1.11;23 ase 'Al'x = 1.21; ase 'Na'x = 1.66;end28 end 57

Listing 12. dist.m% al ulates distan e matrix from either (n,3) or (3n,1) geometry matrix2 fun tion A = dist(x)if size(x,2) == 1 % if (3n,1)n = size(x,1)/3;x = reshape(x,3,n)';7 elsen = size(x,1);endA = zeros(n);for i = 1:n12 for j = (i+1):nA(i,j) = norm(x(i,:)−x(j,:));endendend Listing 13. angle.mfun tion phi = angle(x,a,b, )v1 = x(a,:)−x(b,:);3 v2 = x( ,:)−x(b,:);phi = a os(v1*v2'/norm(v1)/norm(v2));end Listing 14. orre t.m% orre ts for full angle displa ementsfun tion dq = orre t(dq)while (min(dq)<−3 || max(dq)>3)5 dq(dq>3) = dq(dq>3)−pi;dq(dq<−3) = dq(dq<−3)+pi;endend Listing 15. rms.mfun tion x = rms(A)2 x = sqrt(sum(sum(A.^2))/numel(A));end Listing 16. writeX.m% writes artesian de�nition of geometry2 fun tion writeX(x,atoms,�lename)n = length(atoms);�d = fopen(�lename,'w');fprintf(�d,'%d\n reated by /home/hermann/optimization/berny/writeX.m\n',n);7 for i=1:n 58

fprintf(�d,'%−2s %10.5f%10.5f%10.5f\n',atoms{i},x(i,:));endf lose(�d);end Listing 17. readX.m% reads artesian de�nition of geometryfun tion [x,atoms℄ = readX(�lename)4 �d=fopen(�lename,'r');n=fs anf(�d,'%d',1);fgetl(�d);fgetl(�d);x = zeros(n,3);9 for i=1:natoms{i}=fs anf(�d,'%s',1);x(i,:)=fs anf(�d,'%g',3)';endf lose(�d);14 end Listing 18. �leread.m1 fun tion str = �leread(�lename)�d=fopen(�lename,'r');str=fread(�d,'* har')';f lose(�d);end

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