spin-flip

A simple density functional fractional occupation number procedureto determine the low energy transition region of spin-flip reactions

Shu-Guang Wanga�

Workgroup of Theoretical Chemistry, School of Chemistry and Chemical Technology, Shanghai Jiao TongUniversity, 200240 Shanghai, China and Arbeitsgruppe Theoretische Chemie, Department ofChemistry, University Siegen, 57068 Siegen, Germany

Xian-Yang ChenWorkgroup of Theoretical Chemistry, School of Chemistry and Chemical Technology, Shanghai Jiao TongUniversity, 200240 Shanghai, China

W. H. Eugen Schwarzb�

Workgroup of Theoretical Chemistry, School of Chemistry and Chemical Technology, Shanghai Jiao TongUniversity, 200240 Shanghai, China and Arbeitsgruppe Theoretische Chemie, Department ofChemistry, University Siegen, 57068 Siegen, Germany

�Received 23 August 2006; accepted 17 January 2007; published online 29 March 2007�

A computationally simple three-step procedure to survey the energy landscape and to determine themolecular transition structure and activation energy at the intersection of two weakly coupledelectronic potential energy surfaces of different symmetry is suggested. Only commercial softwareis needed to obtain the transition states of, for instance, spin-flip reactions. The computationalexpense is only two to three times larger than that of the standard determination of an adiabaticreaction path. First, the structures of the two electronic initial and final states along a chosen reactioncoordinate are individually optimized. At the “projected crossing,” the two states have the sameenergy at the same value of the reaction coordinate, but different state-optimized partial structures.Second, the unique optimized structure of a low energy crossing point between the two states isdetermined with the help of the density functional fractional occupation number approach. Finally,the respective energy of the two states at the crossing is estimated by a single point calculation. Theprescription is successfully applied to some simple topical examples from organic and frominorganic chemistry, respectively, concerning the spin-flip reactions 3H3CS+→ 1H2CSH+ and7MoCO2→ 5MoCO2→ 3OMoCO. © 2007 American Institute of Physics.�DOI: 10.1063/1.2566404�

I. INTRODUCTION

Elementary chemical reaction steps of a N-atomic sys-tem �if viewed as semiclassical phenomena, and dependingon the chosen approximate Born-Oppenheimer Hamiltonian�occur on a single adiabatic Potential energy hyper surface�PES�, or by transition between two PESs.1 The determina-tion of the reaction path from the educt to the product struc-ture over the transition point on one adiabatic PES isachieved by well documented procedures.2 However, manyimportant reactions occur on several PESs, comprising a va-riety of cases, which still require formidable theoretical andcomputational efforts.2,3

Let A and B denote different electronic space or spinsymmetries. Processes of type 2A→1A occur efficientlythrough conical intersections on a 3N-8 dimensional connec-tion seam of two PESs of same symmetry. Processes of type1B→1A occur by nonadiabatic coupling over a 3N-7 dimen-sional crossing seam of two PESs of different symmetries.The latter description is particularly suited for the discussionof spin-flip reactions of lighter-atomic molecules, where

spin-orbit coupling is the nonadiabatic perturbation.4,5 Reac-tions of type 1 3A→1 1A or 2 1A→1 3A→1 1A have becomeof topical interest. Low energy crossing �LEC� regions oftwo PESs of different spin or position symmetry may easilybe reached by thermal activation, and that may outbalance alow nonadiabatic transition probability �see Fig. 1�.

The common procedure3 is to simultaneously calculatethe two energies of different electronic position and/or spinsymmetries for a given molecular structure and the two gra-

a�Electronic mail: [email protected]�Electronic mail: [email protected]

FIG. 1. Schematic PES �energy of two states S1 and T1 of different elec-tronic symmetries versus reaction coordinate�. �a� and �b� Transitions withhigh Boltzmann and low nonadiabatic transition factors. �c� Transition withlow Boltzmann and high adiabatic transition factor.

THE JOURNAL OF CHEMICAL PHYSICS 126, 124109 �2007�

0021-9606/2007/126�12�/124109/8/$23.00 © 2007 American Institute of Physics126, 124109-1

Downloaded 15 Jul 2009 to 202.120.51.52. Redistribution subject to AIP license or copyright; see http://jcp.aip.org/jcp/copyright.jsp

http://dx.doi.org/10.1063/1.2566404

http://dx.doi.org/10.1063/1.2566404

http://dx.doi.org/10.1063/1.2566404

dients �and in some approaches even the two Hessians�. Thenthese output data are explicitly processed by an interfacedsubroutine, which approaches the minimum energy crossingpoint �MECP� step by step. We here suggest, as an alterna-tive, a technically simple one-PES approach to determine aLEC point on the crossing seam of the two PESs of differentsymmetry, as a good approximation to the MECP. The ap-proximate density functional �DF� self-consistent field �SCF�procedure with fractional orbital occupation numbers�FONs�, as implemented in some commercial software, issufficient for most cases. Any quantum chemical procedurehas its accuracy range, and the concept of a classical trajec-tory over a transition point is an approximate model, any-how. The DF-FON procedure, however, may be replaced byan optimized average level �OAL� multiconfiguration �MC�or complete active space �CAS� procedure. This is advisable,in particular, for higher excited states. If, in the case of smallmodel systems, one wants to improve on the suggested DFT-FON or OAL-MC result with the help of more sophisticatedpost-SCF methods, one can finally use a two-PES-gradient�+Hessian� procedure for the last iteration step, such as oneof those mentioned in Ref. 3.

We will here apply the new procedure at the DF-FONlevel to two spin-flip reactions discussed in the literature,namely, to the 3H3CS+ decay6 and to the 2S+1MoCO2

rearrangement,7 to demonstrate its efficiency.

II. THEORY

A general nonlinear molecule of N atoms has 3N-6=sgeometric nuclear structure degrees of freedom�x ,x2 , . . . ,xs�. The Born-Oppenheimer potential energies En

of the molecular electronic ground and excited states withn=0 and n=1, respectively, form s-dimensional surfaces inthe s+1-dimensional space �E ,x ,x2 , . . . ,xs�. In the case ofdifferent electronic position-spin symmetries, the two PESsE0 and E1 may cross in an s−1 �3N-7� dimensional subspaceof structures. If not prevented by kinematical restraints, theelectronic symmetry changing reaction should occur withsufficient probability, if the nuclear wave packet traverses alow-energy region on that crossing seam.

For a simple example, let us assume two orbitals a ,bforming two-electron states with energies 1A�aa�� 3B�ab�� 1B�ab+ba� in the transition region. In the DF framework,the orbital energies � of the single determinant states 1A and3B obey Janak’s theorem,8,9 which states that for groundstates, all occupied spin-orbitals are below the virtual ones,here,

for 1A�aa�:

��fully occ� � ��a� � ��a� � ��b� � ��b� � ��empty� ,

for 3B�ab�:

��fully occ� � ��a� � ��b� � ��a� � ��b� � ��empty� .

�The simple monodeterminantal description cannot be ap-

plied to the 1B�ab+ba� state. The common DF approach formultideterminantal open shell states, where the energy is rep-

resented as E�1B�=2E�ab�−E�ab�, seems not to be very suit-able for the following prescription aiming at simplicity. Itmust be admitted that such open shell states play an impor-tant role in many photochemical reactions of transition metalcomplexes. A limited MC approach, however, would be ap-plicable.�

In the framework of the FON approach of DF theory,9–11

electronic charge interactions are simulated by an ensembledensity of a mixture of different electronic states of samesymmetry. We here apply the same approach to the mixing oftwo states of different symmetries, to simulate the nonadia-batic transition, as already suggested earlier in similar con-texts, e.g., Ref. 3�o�. The Born-Oppenheimer energy surfaceof intermediate symmetry is now a function of the s geomet-ric structural parameters �x ,x2 , . . . ,xs� and of the electronicmixing parameter n, say, n=sin2 �, 1−n=cos2 �,with the density corresponding to ��n�=cos2� ·��1A�aa��+sin2� ·��3B�ab��. However, for the lowest energy state Jan-ak’s theorem imposes a constraint. While all the lower, fullyoccupied spin-orbitals below a, and all the higher, com-

pletely empty spin-orbitals above b have �nearly always� dif-ferent orbital energies, the two fractionally occupied orbitals

a1−n and bn between a and b must adopt equal orbital ener-gies �a=�b �within the Kohn-Sham spin-symmetry-unrestricted FON-SCF approach�. Therefore the lowestmixed-ensemble energy E�x ,x2 , . . . ,xs ,n� still forms an s di-mensional surface, where for any given x, the optimized par-tial structure �x2 , . . . ,xs� is restricted by the constraint �a

=�b and thereby coupled to the parameter n��0,1�. As longas one of the two states is significantly below the other one,the occupation scheme is pure, n=0 or n=1. But in the cross-

ing region, with EA�EB, n of the mixed state with energy Evaries between n=0 and n=1 �rather abruptly if no explicitcoupling between the two states is included in the Hamil-tonian�, see Fig. 2. In more than half a dozen of cases treatedso far there always occurred a crossing of the two states, ofthe kind E�1A�aa��=E�3B�ab�� with fractionally occupiedorbital energies �a=�b for some intermediate n�0.5. Therespective procedure in the DF-SCF �or similarly in the MC-SCF� framework consists of the following three steps.

In the first step, the initial and final states of the spin-flipreaction under discussion, 1A↔ 3B, are optimized, i.e., thetwo energies En and the s respective structure parameters

FIG. 2. 1A-3B ensemble with fractional occupation n� �0,1�, coupled tooptimized partial structure �xi

n�, i=2 to s, vs reaction coordinate x� �x0 ,x1�. Optimized low energy crossing �LEC� at xpc and �xi

n�opt�� forn�opt�.

124109-2 Wang, Chen, and Eugen Schwarz J. Chem. Phys. 126, 124109 �2007�


Administrator

文本高亮工具

Administrator

文本高亮工具

Administrator

文本高亮工具

Administrator

文本高亮工具

Administrator

文本高亮工具

�xn ,x2n , . . . ,xs

n� are determined for n=0 and n=1, applyingcommon quantum chemical software. Some structural pa-rameter x that varies significantly from the initial state �x=x0� to the final state �x=x1� is then chosen as the “reactioncoordinate.” Of course, there are in general several optionsfor a reasonable choice of x, for instance, an internucleardistance Z1−Z2 or a respective angle Z1−Zj −Z2. However,the finally obtained LEC region with intermediate orbital oc-cupation n� 1

2 at �xpc ,x2pc , . . . ,xs

pc ,n� should be practicallyindependent of that choice. This is corroborated below byour numerical results. The chosen x is then varied and thes−1 partial structure parameters �x2 , . . . ,xs�, also varyingsomewhat during the reaction, are reoptimized in the stan-dard way. The scan of x yields two energy troughs, one forE1A�x ,x2

0 , . . . ,xs0 ,n=0� going up from the minimum at x0

�full line in Fig. 3�, one for E3B�x ,x21 , . . . ,xs

1 ,n=1� going upfrom the minimum at x1 �dashed line�. There is a “projectedcrossing point” xpc with E1A�xpc ,x2

0 , . . . ,xs0 ,n=0� =

E3B�xpc ,x21 , . . . ,xs

1 ,n=1�. The two electronic states have thesame, comparatively low, energy at xpc, but less or moredifferent partial structures �x2

0 , . . . ,xs0� and �x2

1 , . . . ,xs1�. Dif-

ferent cases are also possible for xpc: in many cases, xpc maylie between x0 and x1, as in Figs. 1 and 2 or 4 and 5 below,or it may lie on the same side as in the case of Fig. 3. Belowwe present numerical examples for the different cases.

In the second step, the FON n is varied for fixed xpc

around n=0.5, and the partial structure �x2n , . . . ,xs

n� between�x2

0 , . . . ,xs0� and �x2

1 , . . . ,xs1� is optimized. We change from

constrained minimum E1A�aa��xpc ,x20 , . . . ,xs

0 ,n=0� with ��a��b� to constrained minimum E3B�ab��xpc ,x2

1 , . . . ,xs1 ,n=1�

with ��b��a� over a pass near n=0.5. The energy E of themixed ensemble �1−n� �1A�aa��+n �3B�ab�� takes its maxi-mum value for the energy-minimizing partial structure�x2

pc , . . . ,xspc� with ��a�=��b�. We note that ��a��b� for

n= 12 , or n� 1

2 for ��a�=��b�. In principle one should searchfor the mini-max energy saddle point in the �s+1� dimen-

sional �x ,x2 , . . . ,xs ,n� space under the constraint ��a�=��b�.However, using standard programs such as ADF �Ref. 12� orTURBOMOLE �Ref. 13�, it is much easier to search for theunconstrained stationary energy point in the �s−1� dimen-sional �x2 , . . . ,xs , � space at x=xpc. This is the first basic ap-proximation of the suggested procedure, namely, that xpc isnot reoptimized for �x2

pc , . . . ,xspc�.

In the third step, the FON approach optimizes the orbit-als and the partial structure for the state ensemble. The indi-vidual pure-state energies E1A and E3B at the LEC point�xpc ,x2

pc , . . . ,xspc� are finally calculated. The two states have

slightly different orbital sets with slightly different energies.The same problem occurs for OAL CAS computations. Oursecond approximation is that E1A ,E3B, and the energy at theMECP are not exactly equal. In all the numerical examples

FIG. 4. The 3H3CS+→ 1H2CSH+ rearrangement. Top: Rearrangementscheme with molecular formulas and electronic and geometric symmetrysymbols. Middle: Optimized energy curves E along reaction coordinate �=��H1CS� for 3A of H3CS+ �full line� and 1A of H2CSH+ �dashed line�.Bottom: Ensemble energy E and orbital energies �i of 6a�1−n and 7a�n formixed ensemble �1−n��1A�+n�3A�, when varying the FON n, with opti-mized partial structure �xi

n�. Energies in kJ/mol.

124109-3 Simple DF-FON procedure J. Chem. Phys. 126, 124109 �2007�


presented below the deviations E1A�E3B�EMECP turned outto be smaller than the reliability of the applied commonquantum chemical procedures.

Summarizing the proposed prescription goes as follows:First scan the reaction coordinate x and optimize the twopartial structures for electronic states 1A and 3B by twosingle state energy minimizations for fixed n=0 and 1 andget the xpc. Then fix x=xpc and scan n, optimizing a singlepartial structure, and get the LEC point. Finally, calculate theenergy at the LEC as E1A�E3B. In the next section we willpresent two molecular examples and investigate how well

the relations E�E1A�E3B�EMECP and ��a��b� are ful-filled, how strongly the results depend on the choice of thereaction coordinate, and how well the results agree with re-sults from more sophisticated literature computations.

III. APPLICATIONS

We investigate the following two topical examples: �i�one from organic chemistry, the H2 elimination from 3H3CE+

yielding 1H2+ 1HCE+ �here for the case of E=sulfur�; �ii�one from transition metal catalysis, the rearrangement from7Mo+CO2 to 3MoO+CO. In both arbitrarily selected caseswe obtain satisfactory results. We use the ADF2004 programsuite12 and also GAUSSIAN03.14 Several different DFs wereapplied such as BP, PW, BLYP, or B3LYP.15 We apply theextended triple-�-polarized Slater and GAUSSIAN basissets, respectively, from the literature.16

A. The thiomethoxy cation

The chemistry of methoxy-type triplet cations has beendiscussed in detail by Aschi et al.6,17 One of the paths of H2

elimination from 3H3CS+ starts as sketched in Fig. 4 �top�.

The calculated structure parameters and energies of the ini-tial 3A1 state, of the final 1A� state, and of the mixed en-semble at the LEC for two different reaction coordinates, x=R�S–H1� and x=� SCH1, are presented in Table I andcompared with literature values �in parentheses�.6

Initial and final states. At first we estimated the reliabil-ity of the results of the stationary initial and final states.Different extended basis sets �STO in ADF, GTO in GAUSS-IAN� and different optimization procedures result for thebond lengths �in picometer� and angles �in degree� in some-what different first digits after the decimal points. The non-bonded distances and the structure parameters of the moreflexible transition states may be uncertain by more than 1 pmor 1°. The results from BLYP are displayed in the first andlast columns of Table I �with the literature values of Aschi etal.17 in parentheses�. Using different density functionals ordifferent ab initio post-SCF approaches, larger differencesare obtained, in particular, concerning the energies. We men-tion an extreme example. The 3H3CS– 1H2CSH energy dif-ference from photoionization mass spectrometry18 is−140±8 kJ/mol. With ADF-BLYP we obtain −156 kJ/mol�Table I, lowest row�. On the other hand, several other ADF-DFs and all GAUSSIAN-DFs as well as CC-SD�T� and MP4yield low values around −110 to −120 kJ/mol. Using verylarge basis sets, the upper triplet is stabilized by a few kJ/molfor all methods. The ADF-BLYP singlet-triplet distance sta-bilizes near 150 kJ/mol, which is not in real contradiction tothe experimental value.

The LEC states. The LEC was determined in differentways. The values in the left LEC column of Table I areobtained for the SCH1 angle as the reaction coordinate �Fig.4�, with three different criteria: The left entries refer to thecrossing of the two partially occupied orbital energies

TABLE I. The 3H3CS+→ 1H2CSH+ rearrangement. Structural parameters of initial 3A1 state, final 1A� state andof the 3A− 1A ensemble state at the LEC �from BLYP functional with Slater VTZ2P basis�. Distances in pm,angles in deg, and energies in kJ/mol.

Reaction coordinateCriterion

Initial 3A1 �C3v�a LEC 3−1A �C1�x=� SCH1

��6a�=��7a� / E max/E3A=E1A

LEC� 3−1A �C1�a

x=R�S–H1�E max

Final 1A� �Cs�a

R�C–H3� �pm� 109.5 �110.1� 108.3/108.2/108.3 108.3�109.2� 107.7 �108.8�R�C–H2� �pm� 109.5 �110.1� 108.3/108.2/108.3 108.3�109.2� 107.5 �108.7�R�C–H1� �pm� 109.5 �110.1� 116.8/117.0/116.8 115.1 �122.6� 238.3 �227.3�R�S−H1� �pm� 232.5 �235.6� 179.1/179.3/179.5 177.9 �175.8� 135.5 �136.1�R�C–S� �pm� 172.3 �175.1� 171.6/172.1/171.5 168.8 �169.0� 161.8 �162.6�� SCH3 �deg� 109.4 �109.3� 118.3/118.1/118.3 118.6 �119.7� 117.3 �117.3�� SCH2 �deg� 109.4 �109.3� 118.3/118.1/118.3 118.6 �119.7� 123.2 �123.4�� SCH1 �deg� 109.4 �109.3� 74.1/73.9/74.3 74.9 �72.2� 36.7 �36.3�� SCH1 �deg� 26.3 �26.2� 38.9/41.3/38.8 38.7 �41.6� 97.8 �98.7�

n for ��6a�=��7a� 0.500/0.486/0.500 0.481E�3A�=E�1A� at xpc

¯/¯/71.9 71.8

E at the LEC 80.2/80.5/79.8 80.1�106.5/*99.1�

E�3A� at stable initialand LEC structures

0 79.0/79.2/78.0 78.6

E�1A� at stable finaland LEC structures

77.7/78.2/78.0 77.6 −156.2�−109.5/*−119.6�

aIn parentheses: literature values �Ref. 6� of the initial and final states and the MECP �B3LYP/*CC-SD�T� with6-311+G�d , p� /pol-VTZ bases�.



Administrator

文本高亮工具

��6a��=��7a��, the middle entries to the maximum of the

ensemble energy E, and the right entries to the crossing ofthe energies of the two pure states 3A�H3CS+� and1A�H2CSH+�. The right LEC� column, for the internuclearSH1 distance as the reaction coordinate, contains the resultsfrom maximizing the ensemble energy. The values in paren-theses are the results of Aschi et al.17 for the MECP.

The uncertainties of the results for the LEC and for thestationary states are similar. Most DFs �not displayed here�and ab initio approaches give barrier heights between 100and 115 kJ/mol, except for the 80 kJ/mol by ADF-BLYP�Table I�. This deviation for the triplet-educt-singlet-productbarrier is about one-half of the ADF-BLYP deviation for thetotal educt-product energy distance mentioned above �about40 kJ/mol�. Different reaction coordinates and different

search criteria, that is, ��a�=��b� or E mini-max or EA=EB

or n=0.5, yield only slightly different LEC results, usuallywithin about 1 pm, 1°, and 3 kJ/mol. The individual 1A and3A states differ by 0–1 kJ/mol at the LEC, which is a toler-able uncertainty for the crossing point. To obtain the chemi-cally relevant activation barrier, the calculated78–79 kJ/mol of E�1A��E�3A� at the LEC �from ADF-BLYP� are to be reduced by a zero point vibrational energycorrection �ZPE� of 8–10 kJ/mol.

The energy of the individually orbital- and structure-optimized states at xpc is E�1A��at �xpc ,xi

0�� =E�3A� �at�xpc ,xi

1�� = 72 kJ/mol. The energy of 1A and of 3Aat the unique LEC structure �i.e., at �xpc ,xi

0.5�� is 6 kJ/mol

higher, i.e., 78 kJ/mol. The FON ensemble energy E at theLEC with a single ensemble orbital set is even higher, here80 kJ/mol, i.e., only another 2 kJ/mol.

B. The MoCO2 system

Souter and Andrews reported the reaction of �7S�Moground state atoms with �1�CO2 molecules to form a�3A��OMoCO product.7 The respective reaction steps arebound to involve transitions between multielectronic statesof different spin multiplicities, namely, two spin flips be-tween septet and quintet �LEC-1�, and between quintet andtriplet �LEC-2�, and in between one adiabatic transition onthe quintet PES, see Fig. 5. The spin-orbit coupling of Mo, amiddle-heavy atom, is of the order of 10 kJ/mol, which sup-ports spin-flip transition probabilities. We used the PW91DF.

Septet-pentet transition (Table II). The first LEC-1 of theMoCO2 system corresponds to a “continuous reaction,”where R�Mo–C�, R�Mo–O1�, and the OCO angle steadilydecrease, and R�C–O1� steadily increases, from the 7Aground state of Mo+CO2 with reference energy E=0, to 7Aof MoCO2 at xpc �+11.9 kJ/mol�, to the 7–5A ensemble statebarrier at the LEC-1 �+43 kJ/mol�, to 5A at xpc �again+11.9 kJ/mol�, and to the secondary stationary energy mini-mum of 5A�MoOCO� at E=−45.6 kJ/mol. R�C–O2� variesonly a little. The individually optimized partial structures�xi

0� and �xi1� of the two states 7A and 5A at the projected

crossing are remarkably different, and the LEC structurewith �xi

n� is not nearly the arithmetic average, see the two columns of Table II. Correspondingly, the optimized connec-

tion between 7A�xpc ,xi0� and 5A�xpc ,xi

1�, i.e., the ensemble

energy curve E in Fig. 5 �bottom�, is not a symmetric pa-rabola over a straight line at xpc, as in the sketch of Fig. 3.Correspondingly, the optimized n value is off 0.5, namely,about 0.6. Again the two spin states agree within 1 kJ/mol at�xpc ,xi

0.6�, E�7A��E�5A��46 kJ/mol �to be reduced by3–4 kJ/mol due to the ZPE�. Because of the large structuraldifferences between 7A�xpc ,xi

0�, 7–5A�xpc ,xi0.6�, and

FIG. 5. The 7Mo+CO2→ 5MoOCO→ 5OMoCO→ 3OMoCO rearrange-ment. Top: Rearrangement scheme with molecular formulas and electronicsymmetries and energies. Middle: Optimized energy curves around LEC 1along the reaction coordinate �R�Mo–C� in pm� for �7A�MoCO2 �dashed

line� and �5A�MoOCO �full line�. Bottom: 7−5A ensemble energy E andpartially occupied frontier orbital energies � j when varying the FON n, withoptimized partial structure �xi

n�. Energies in kJ/mol.



5A�xpc ,xi1�, the barrier value is 34 kJ/mol above the

individually optimized states at xpc, E�7A�xpc ,xi0�� =

E�5A�xpc ,xi1�� = 12 kJ/mol. In the present case, the ensemble

energy E of 43 kJ/mol happens to be 3 kJ/mol below thebarrier between the individual states.

Pentet-triplet transition of OMoCO (Table III). Thiselectronic-nuclear rearrangement is again investigated withtwo different reaction coordinates, the CMoO angle and theCO distance. The reaction coordinates around the LEC-2 areof U-turn type: Both R�C–O� and � �CMoO�, and also �

�MoCO��, at first increase and then decrease significantly�Fig. 3 above�. Our procedure also works in this case. For thetwo differently chosen reaction coordinates, the LEC struc-tures differ by � �Table III�: a few picometer or degree, withbarrier energies differing by only 1–2 kJ/mol. It means thatthe crossing seam is rather flat at the LEC and a sharp MECPhas not much physical meaning. The distance from the pro-jected crossing energies E�3A�=E�5A� of 2 1

2 or 4 kJ/mol tothe LEC barrier at E�3A��E�5A��2.8 or 5 kJ/mol is also

very small. This time the ensemble energy E is nearly10 kJ/mol above the barrier.

IV. CONCLUSIONS

The LEC region of electronic states A, B of differentelectronic symmetries, with differently occupied spin-orbitals a, a, b, in an s dimensional space of geometric struc-ture degrees of freedom is important for many nonadiabaticphotochemical, photobiochemical, and catalytic reactions.The proposed procedure surveys the reaction path on therelevant energy landscape in three steps: �i� up a valley onthe PES of initial electronic symmetry, �ii� symmetry changeover the s−1 dimensional crossing seam at low energy, and�iii� finally down in another PES valley to the final state. Thisis technically achieved with common commercial softwareof computational chemistry, without any additional program-ming work. To this aim, three coordinate scans are per-formed.

�1� Scanning a chosen A�0�← �B��1� reaction coordinatex from x1 up to xpc, for fixed occupation number n=1, i.e.,from the initial state 3B�n=1� with structure �x1 ,x2

1 , . . . ,xs1�

up to the low energy projected crossing, corresponding to �i�.�2� Scanning x from x0 up to xpc, for fixed n=0, i.e.,

TABLE II. The 7Mo+CO2→ 5MoCO2 rearrangement. From PW91 DF with scalar relativistic correction andTZP basis. Distances in pm, angles in deg, and energies in kJ/mol. is the difference between the 7A or 5Astates with partially optimized structures at xpc and the 7−5A ensemble state with LEC structure.

x=R�Mo–C�Initial

�7S�Mo+CO2

�7A�MoCO2

at xpc LEC-1

�5A�MoCO2

at xpc�5A�MoCO2

secondary min.

R�C–O1� 117.3 119.7 3.8 123.5 6.4 129.9 136.3R�C–O2� 117.3 119.7 −0.8 119.1 0.4 119.5 120.0R�Mo–C� � xpc=249.6 200.7� OCO 180 159.6 −7.2 152.4 −13.6 138.8 133.9

� MoCO ¯ 100.2 −31.0 69.2 −12.9 56.3 68.9n 1 1 0.5986 0 0

E�7A� , E�7−5A� ,E�5A�at LEC �xpc ,xi

0.6�+45.8 +43.0 +46.3

Individually optimized Eof 7A or 5A

0 +11.9 +11.9 −45.6

TABLE III. The 5OMoCO→ 3OMoCO rearrangement. Structural parameters from PW91 functional with TZPbasis and scalar relativistic correction: distances in pm, angles in deg, and energies in kJ/mol. means thedifference between the joint LEC ensemble structure and the stationary structures of the quintet and triplet staes.� represents the difference of the two LEC from the two different reaction coordinates.

Reaction coordinateCriterion

�5A�OMoCO LEC-2� �CMoO�

EA=EB /��a�=��b�

� LEC-2�R�C–O�EA=EB

�3A�OMoCO

R�Mo–O� 173.0 Small 172.8/172.2 1 12 173.9 −3 170.0

R�Mo–C� 209.3 208.0/208.2 7 214.9 −17 194.8R�C–O�� 116.0 Small 116.3/116.3 0.3 116.0 +1 117.2

� �MoCO�� 169.4 +5 176.1/175.9 3 173.0 −4 170.9� �CMoO� 123.8 +9 135.2/135.2 4 131.1 −29 104.2

R�C–O� 337.7 +16 352.3/352.0 2 354.4 −65 288.3n for ��6a�=��7a� 0.5/0.4943 0.4886

E�3A� or E�5A�of GS or at xpc

0 4.1 1 12 2.6 −49.3

E at LEC 14.3/14.7 2 12 12.0

E�5A� at LEC 5.2/5.4 2 12 2.8

E�3A� at LEC 4.4/4.9 2 2.8



from the final state 1A�n=0� with structure �x0 ,x20 , . . . ,xs

0� upto the projected crossing, corresponding to �iii�.

�3� Scanning the incoherent 3B− 1A ensemble mixingaround n=0.5, for x fixed at xpc, by optimizing the structure�xpc ,x2

pc , . . . ,xspc� on the crossing seam, corresponding to �ii�.

Concerning point �ii�, i.e., �3�, the DFT-FON approxima-tion is applicable to all cases, where the electronic states canbe assigned to single main configurations. For multiconfigu-rational excited states, the average state MC approach maybe applied for the incoherent mixing of different symmetries.�Usually these techniques are used for coherent mixing oftwo functions of same symmetry, such as those which areconnected by an s−2 conical intersection seam.� When thelow energy ensemble structure on the s−1 dimensionalcrossing seam has been determined, the individual state en-ergies E�3B��E�1A� at the LEC are afterwards determinedby single point calculations. Finally, the nonadiabatic transi-tion probability may be estimated by Landau-Zener-type ap-proximations.

The approach to determine the LEC structure and energybarrier has been tested for several different topical cases ofearly and late downhill type, with S and U shape reactionpaths, over more or less flat crossing seams. The obtainedstructural LEC parameters lie in the reliability ranges of theapplied computational procedures for stationary states �be itDFT or common ab initio post-HF CAS-PT or CC-SDT�.The DFT values obtained here agree reasonably with thoseof more sophisticated literature values, i.e., within the orderof picometer and degree. In the case of floppy structures theuncertainty may be somewhat larger.

The needed CPU time is comparable to common adia-batic reaction path determinations. The problem of findingthe transition state is similar in both cases and sometimesneeds some skill. At first one must find the initial and finalstates of the reaction. From two additional, sensibly chosenintermediate points one obtains an approximation to xpc,which is usually refined with three to five steps. The samenumber of steps is usually sufficient for the FON optimiza-tion. For a small system such as CH3S+, we need about 1

2 hCPU �1-processor 3 GHz personal computer�.

The main problem seems not to be connected with theapproximations inherent in the suggested search procedure,but in the reliability of molecular energies of open shellstates and of transition states obtained from common quan-tum chemical programs. The prescription, which models anapproximate reaction path, will facilitate the search andphysicochemical understanding of nonadiabatic transitionsover LEC seams. The chemical implications of the presentedand further LEC point determinations will be published else-where.

ACKNOWLEDGMENTS

The authors thank J. N. Harvey, M. Holthausen, H. Köp-pel, M. Lundberg, and S. Shaik for constructive commentson the manuscript. They acknowledge financial support bythe National Natural Science Foundation of China �No.20373041 and 20573074� and by the Fonds der ChemischenIndustrie �Grant No. 160273�. One of the authors �W.H.E.S.�

acknowledges hospitality and financial support of SJTU. An-other author �S.G.W.� acknowledges hospitality of USiegenand financial support from DAAD.

1 H. Hellmann and J. K. Syrkin, Acta Physicochim. URSS 2, 433 �1935�;M. Born and K. Huang, Dynamical Theory of Crystal Lattices �Claren-don, Oxford, 1966�.

2 H. B. Schlegel, in Modern Electronic Structure Theory, edited by D. R.Yarkony �World Scientific, Singapore, 1995�, Vol. 1, p. 459; Adv. Chem.Phys. 67, 250 �1987�; D. R. Yarkony, Rev. Mod. Phys. 68, 985 �1996�.

3 �a� W. Domcke, D. R. Yarkony, and H. Koeppel, Conical Intersections:Electronic Structure, Dynamics and Spectroscopy �World Scientific, Sin-gapore, 2004�; �b� J. C. Lorquet, Int. J. Mass. Spectrom. 200, 43 �2000�;�c� W. H. Miller, N. C. Handy, and J. E. Adams, J. Chem. Phys. 72, 99�1980�; �d� N. Koga and K. Morokuma, Chem. Phys. Lett. 119, 371�1985�; �e� D. R. Yarkony, J. Phys. Chem. 97, 4407 �1993�; �f� M. J.Bearpark, M. A. Robb, and H. B. Schlegel, Chem. Phys. Lett. 223, 269�1994�; �g� D. R. Yarkony, in Modern Electronic Structure Theory, editedby D. R. Yarkony �World Scientific, Singapore, 1995�, Vol. 1, p. 642; �h�Q. Cui and K. Morokuma, Chem. Phys. Lett. 272, 319 �1997�; �i� N.Harvey, M. Aschi, H. Schwarz, and W. Koch, Theor. Chem. Acc. 9, 95�1998�; E. L. Øiestad, J. N. Harvey, and E. Uggerud, J. Phys. Chem. A104, 8382 �2000�; �j� T. Chachiyo and J. H. Rodrigues, J. Chem. Phys.123, 094711 �2005�; �k� M. Lundberg and P. E. M. Siegbahn, J. Phys.Chem. B 109, 10513 �2005�; �l� H. Schwarz, Int. J. Mass. Spectrom.237, 75 �2004�; �m� J. Schoeneboom and W. Thiel, J. Am. Chem. Soc.126, 4017 �2004�; �n� M. Barbatti, M. Ruckenbauer, and H. Lischka, J.Chem. Phys. 122, 174307 �2005�; �o� G. Granucci, M. Persico, and A.Toniolo, ibid. 114, 10608 �2001�; �p� A. Farazdel and M. Dupuis, J.Comput. Chem. 12, 276 �1991�; �q� L. De Vico, M. Olivucci, and R.Lindh, J. Chem. Theory Comput. 1, 1029 �2005�; �r� F. Jensen, J. Am.Chem. Soc. 114, 1596 �1992�; J. Chem. Phys. 119, 8804 �2003�; �s� J.M. Anglada and J. M. Bofill, J. Comput. Chem. 18, 992 �1997�; �t� M.Baer, Electronic Nonadiabatic Coupling Terms and Conical Intersections�Wiley-VCH, Weinheim, 2006�.

4 S. Shaik, J. Am. Chem. Soc. 100, 18 �1978�; 101, 2736 �1979�; 102, 122�1980�; D. Danovich and S. Shaik, ibid. 119, 1773 �1997�; D. Schröder,S. Shaik, and H. Schwarz, Acc. Chem. Res. 33, 139 �2000�; D. A. Platt-ner, Angew. Chem., Int. Ed. 38, 82 �1999�.

5 J. N. Harvey, R. Poli, and K. M. Smith, Coord. Chem. Rev. 238/239, 347�2003�; R. Poli and J. N. Harvey, Chem. Soc. Rev. 32, 1 �2003�; R. Poli,J. Organomet. Chem. 689, 4291 �2004�.

6 M. Aschi and F. Grandinetti, J. Chem. Phys. 111, 6759 �1999�, see alsoM. Teruel, P. R. P. de Moraes, L. A. Xavier, and J. M. Riveros, Eur. J.Mass Spectrom. 9, 279 �2003�; J. R. Flores, C. Barrientos, and A. Largo,J. Phys. Chem. 98, 1090 �1994�.

7 P. F. Souter and L. Andrews, Chem. Commun. �Cambridge� 1997, 777�1997�.

8 J. F. Janak, Phys. Rev. 18, 7165 �1978�.9 S. G. Wang and W. H. E. Schwarz, J. Chem. Phys. 105, 4641 �1996�.

10 J. C. Slater, J. B. Mann, T. M. Wilson, and J. H. Wood, Phys. Rev. 184,672 �1969�.

11 P. R. T. Schipper, O. V. Gritsenko, and E. J. Baerends, Theor. Chem. Acc.99, 329 �1998�; M. Filatov, Chem. Phys. Lett. 304, 429 �1999�, J. Phys.Chem. A 104, 6628 �2000�.

12 E. J. Baerends, D. E. Ellis, and P. Ros, Chem. Phys. 2, 41 �1971�; G. teVelde and E. J. Baerends, J. Comput. Phys. 99, 84 �1992�; P. M. Boer-rigter, G. te Velde, and E. J. Baerends, Int. J. Quantum Chem. 33, 87�1988�.

13 R. Ahlrichs and M. von Arnim, in Methods and Techniques in Computa-tional Chemistry: METECC-95, edited by E. Clementi and G. Corongiu�STEP, Cagliari, 1995�, Chap. 13, p. 509.

14 M. J. Frisch, G. W. Trucks, H. B. Schlegel et al., GAUSSIAN03, GaussianInc., Wallingford, CT, 2004.

15 J. C. Slater, Phys. Rev. 81, 385 �1951�; S. H. Vosko, L. Wilk, and M.Nusair, Can. J. Phys. 58, 1200 �1980�; J. P. Perdew, Phys. Rev. B 33,8822 �1986�; 34, 7406 �1986�; A. D. Becke, J. Chem. Phys. 88, 2547�1988�; 98, 5648 �1993�; J. P. Perdew, K. Burke, and Y. Wang, Phys.Rev. B 54, 16533 �1996�; J. P. Perdew, K. Burke, and M. Ernzerhof,Phys. Rev. Lett. 77, 3865 �1996�.



Administrator

文本波浪线工具

16 G. J. Snijders, E. J. Baerends, and P. Vernooijs, At. Data Nucl. DataTables 26, 483 �1982�, Internal Reports �Theoretische Chemie, VrijeUniversiteit Amsterdam, Amsterdam, 1982�; R. Ditchfield, W. J. Hehre,and J. A. Pople, J. Chem. Phys. 54, 724 �1971�.

17 M. Aschi, J. N. Harvey, C. A. Schalley, D. Schröder, and H. Schwarz,Chem. Commun. 1998, 531 �1998�; J. N. Harvey and M. Aschi, Phys.Chem. Chem. Phys. 1, 5555 �1999�.

18 B. Ruscic and J. Berkowitz, J. Chem. Phys. 97, 1818 �1992�.



spin-flip

Documents