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Electrostatic interactions between molecules fromrelaxed one-electron density matrices of the coupledcluster singles and doubles modelTATIANA KORONA a , ROBERT MOSZYNSKI a & BOGUMIL JEZIORSKI aa Department of Chemistry, University of Warsaw, Pasteura 1, 02-093, Warsaw, Poland

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To cite this article: TATIANA KORONA, ROBERT MOSZYNSKI & BOGUMIL JEZIORSKI (2002): Electrostatic interactions betweenmolecules from relaxed one-electron density matrices of the coupled cluster singles and doubles model, Molecular Physics: AnInternational Journal at the Interface Between Chemistry and Physics, 100:11, 1723-1734

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MOLECULAR PHYSICS, 2002, VOL. 100, NO. 1 1, 1723- 1734 + Taylor &Francis 0 Taylor hFrancir Group

Electrostatic interactions between molecules from relaxed one- electron density matrices of the coupled cluster singles and doubles

model TATIANA KORONA, ROBERT MOSZYNSKI and BOGUMIL JEZIORSKI*

Department of Chemistry, University of Warsaw, Pasteura 1, 02-093 Warsaw, Poland

(Received 22 July 2001; accepted 15 October 2001)

The influence of electron correlation on the electrostatic interaction between closed shell molecules is studied using the relaxed electron densities of the coupled cluster singles and doubles (CCSD) model. The corresponding CCSD one-electron density matrices are efficiently computed without full four-index transformation by employing the generalized exchange and Coulomb operator technique. Using several representative van der Waals and hydrogen bonded complexes it was found that in most cases the convergence of the Merller-Plesset expansion of the electrostatic energy, restricted to single, double and quadruple excitations, is satisfactory and the fourth-order triple excitation term is more important than the sum of the fifth- and higher-order contributions from CCSD theory. The importance of the CCSD correlation correction to the electrostatic energy was gauged by comparison of the total inter- action energy computed by symmetry-adapted perturbation theory (SAPT) and by the super- molecular CCSD(T) approach (coupled cluster singles and doubles model with a non-iterative inclusion of triple excitations). Except for the CO and N2 dimers, very good agreement between the two sets of results is observed. For the difficult case of the CO dimer the difference between the SAPT and CCSD(T) results can be explained by the truncation of the SAPT expansion for the dispersion energy at second order in the intramonomer correlation operator.

1. Introduction Electrostatic interactions play an important role in

determining the structure of dimers consisting of polar molecules, in particular of hydrogen bonded complexes [l-31. Often the evaluation of the electrostatic inter- action energy for such systems is performed by approx- imating the electrostatic potential of a molecule by that resulting from a one-centre [4] or a multi-centre distri- bution of multipole moments [5-71. However, the elec- trostatic energy contains also important short range terms due to the mutual penetration (charge overlap) of monomers’ electron clouds. This short range part of the electrostatic energy makes significant contributions to the stabilization energy of atom-molecule van der Waals complexes [8, 91, and cannot be neglected in any accurate calculations of the potential energy sur- faces for such systems. For instance, for the non-polar Ar-CH4 complex the electrostatic energy, which repre- sents a pure penetration effect in this case, represents as much as 42% of the total interaction energy at the mini- mum, and is even larger than the induction energy [9].

* Author for correspondence. e-mail: jeziorsk@chem.uw. edu.pl

Thus, an accurate description of the electrostatic energy is necessary to obtain reliable potential energy surfaces, even for non-polar systems.

In 1993 it was proposed [lo] that the electrostatic energy be calculated from a Mailer-Plesset (MP) expansion in the intramonomer correlation. Two correlation expansions were considered. In the first approach the wavefunctions of the isolated monomers were expanded as perturbation series with respect to the correlation operators of the monomers. A straightfor- ward application of these wavefunction expansions in the expression for the electrostatic energy leads to the so-called non-relaxed Msller-Plesset series for In the second approach an external field dependent Msller-Plesset partitioning of the Hamiltonian is used, cf. [lo], equation (23), which leads to a proper account of the orbital relaxation or orbital response effects [Il, 121. In [lo] the convergence of these two expansions was investigated for simple four-electron dimers: Hez, He- HZ, and (H&, by comparison of the perturbative results through fourth order (in the intramonomer correlation operator) with the full configuration interaction (FCI) results obtained in the same basis set. Both expansions showed satisfactory convergence in fourth-order,

Moleculur Physics ISSN 002G3976 print/ISSN 1362-3028 online :c) 2002 Taylor & Francis Ltd http://www.tandf.co.uk/journals

DOI: 10.1080/00268970110105424

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1724 T. Korona et al.

although the relaxed results were somewhat closer to the FCI data.than the non-relaxed. The results reported in [lo] suggested that an accurate description of the elec- trostatic interactions can be obtained by employing the relaxed Mdler-Plesset expansion, and this approach has been used with success for numerous van der Waals complexes (see, e.g. [8, 9, 131).

Thus it came as a surprise that a similar correlation expansion for the dipole moment is strongly divergent [14, 151. In fact, the results presented by Olsen et al. [14] suggested that the second-order Merller-Plesset result is reliable, while the summation of higher-order terms only worsens the agreement with the FCI results. Since in the multipole approximation [4] the MP expansions of the dipole moment and of the electrostatic energy are closely related, the good convergence towards the FCI results for EL,’,), in [lo] could be fortuitous, as this work concen- trated on restricted basis sets and a rather specific class of model systems, namely four-electron dimers. Thus, the question of the convergence of the MP expansion for the electrostatic energy should be reexamined, and non-perturbative approaches for accurate calculations of this term should be devised.

The aim of the present paper is to investigate the convergence of the M~iller-Plesset expansion of the elec- trostatic energy by comparison with the converged results obtained from the electron densities of the coupled cluster singles and doubles (CCSD) model [ 16191. An efficient implementation of the corre- sponding relaxed (and non-relaxed) one-electron density matrices employing the generalized exchange and Cou- lomb operator technique will be presented and dis- cussed. The role of the triple-excitation terms will be investigated at the fourth-order M~ller-Plesset level. Numerical results will be presented for several represen- tative hydrogen bonded and van der Waals complexes: (H20)2, (HF)2, (C0)2, (N2)2, and He-H20. To put our results in a wider context, also reported in this paper are other components of the interaction energy for these complexes, as defined by symmetry-adapted perturba- tion theory (SAPT) [20]. Finally, the SAPT interaction energies are compared with high level supermolecular results.

2. Many-body theory of electrostatic interactions We consider the interaction of two closed-shell mono-

mers A and B. The energy of the electrostatic interaction between A and B, defined as the first-order correction

energy [20], can be written as Eelst (1) in the polarization expansion of the interaction

= J JpA(rl)pa(rz)u(rl,rz)drldrz, (1)

where pA(r) and pB(r) are the electron densities of the free monomers and the generalized interaction operator u(rl,rz) is defined by [lo], equation (8). In our previous work on the electrostatic interactions [ 101 the electron densities appearing in equation (1) were represented by suitable Merller-Plesset perturbation expansions. This leads to the Marller-Plesset expansion of the electrostatic energy

m

n=O

where denotes the correction to the electrostatic energy which is of the nth order in the intramonomer correlation. The subscript ‘resp’ means that the so-called response or orbital relaxation effects are taken into account [ 1 I]. The explicit expression for is given by:

(3)

X = A or B, denotes the Ith-order Mdler-Plesset correction to the exact electron density obtained by differentiation of the lth-order Marller- Plesset energy of system X in the presence of the per- turbing field b(r) given by the density operator, cf. [lo], equations (20)-(26).

Given the divergent nature of the M~ller-Plesset expansion for the electric dipole moment, one may expect that the expansion (2) will be divergent as well. Therefore, in the present work we shall consider an approximation to obtained by replacing the exact electron densities appearing in equation (1) by the elec- tron densities obtained from the coupled cluster method restricted to single and double excitations (CCSD):

Here pX,resp(r), (0

(CCSD) = J J pzCSD (r I )pgCSD (r2)u(r 1 , r2)drl dr2.

(4)

The CCSD electron densities appearing in equation (4) can be computed without or with the orbital relaxation. In the latter case, the corresponding electrostatic energy will be denoted by Ek:&,,,(CCSD). The CCSD method is known to be correct through the third order in the monomer correlation. Thus, (CCSD) includes all second- and third-order diagrams (in the intramonomer fluctuation potentials) and sums up a specific class of diagrams (connected diagrams resulting from single, double, and disconnected quadruple excitations) to infinite order.

To investigate the convergence of expansion (2) we shall compare the electrostatic energy computed by sum-

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Electrostatic interactions between molecules by CCSD 1725

ming the single-, double-, and quadruple-excitation dia- grams through fourth order:

+ 'L;iiresp(S"Q), (5)

with the converged value of EL;jt,resp (CCSD). Here (SDQ) denotes the fourth-order Merller-Plesset

correction to the electrostatic energy obtained by neglecting the connected triple excitation operator,

(122) + Eelst,resp 1

where ELf,:7Lsp denotes a correlation correction to the electrostatic energy of the nth order in the correlation on A, and of the mth order in the correlation on B. The term Ek;sy:esp (SDQ) (SDQ)) was calculated numerically as a first derivative of the MP4(SDQ) energy of the monomer A(B) in the unperturbed Har- tree-Fock electrostatic field of the monomer B(A), while the last term was obtained from the analytical relaxed MP2 electron densities, available in the Molpro package [21, 221. Thus, the difference,

'!;jt,resp(CCSD) = EL;jt,resp (CCSD) - EL;/[,resp (4; SDQ) 7

(7) represents the contribution of the fifth- and higher-order SDQ diagrams. (Note that the term does not appear in equation (5) since it vanishes due to the Bril- louin theorem.) The . role of the triple-excitation dia- grams will be demonstrated by considering their contribution to Ekfs41iresp, i.e. EL;~~,,,,(T). The correction Ek;;!resp(T) was calculated in the same way as

first cferivative of the fourth-order triple-excitation energy of one monomer perturbed by the Hartree- Fock electrostatic field of the other monomer. Finally, our best approximation to the exact electrostatic inter- action energy will be given by:

Eelst,res (104) (SDQ)+ Eelst,resp (140) (SDQ), i.e. numerically as a

-(1) - (1) (14) 'elst,resp - Ee~st,resp (CCSD) + Eelst,resp(T). (8)

This energy will be compared with the approximation proposed previously in [lo]:

~ L ; j t , ~ ~ ~ ~ (4) = Ek2,resp (4; SDQ) + ELt,41!resp (TI. (9)

It is worth noting that for polar systems the sum Eelst,resp ( 1 4 + EL;:$p, n 5 4, determines the long range be- haviour of the MPn su ermolecular interaction energy

supermolecular MP5 interaction energy [23]. Thus, the [lo], while the term Ee,st,resp $22) appears as a part of the

CCSD electrostatic energy will govern the long range behaviour of the supermolecular CCSD interaction energy, but also will account for some important dia- grams of the CCSDT(Qf) theory [24]. These diagrams are not correctly included in the CCSD(T) or CCSDT theories, and may be responsible for the difficulties these theories encounter in describing the interaction in the CO dimer [23, 25, 261.

3. cluster theory

We shall follow here the basis set independent approach of Bukowski et al. [27]. A straightforward extension of this approach to the coupled cluster singles and doubles level leads to the following expression for the relaxed first-order property Eresp corresponding to the perturbation (external field) U:

Relaxed one-electron density matrix in coupled

( 1 )

E$;B, = (ePT(u + [ H , c1 - c,l)eT) t

(GTl[ePTHeT,T(')] +epT(u + [ H , C, - cl])eT) t

+ ( [ e ~ ~ ~ e ~ , ~ ( l ) ] ) ,

(10) where T ( ' ) is the first-order cluster operator satisfying

= 0, (11)

His the Hamiltonian of the system without the external field, T is the unperturbed cluster operator, and Cl is the CHF operator defined by the equation

(12) t (67-1 I[H, CI - C,] + U ) = 0.

By ( X ) we denote the expectation value ( @ l X @ ) of an operator X with the field independent Hartree-Fock determinant @, (XIY) is the shorthand notation for (X@lY@) , and 6T is an arbitrary variation of T (TI is the one-particle part of 7). Note that equations (10) and (1 1) can be obtained from the non-relaxed theory of Monkhorst [28] by the substitution

(13) t u+u + [H, c, - C,].

Equations (1 0 ) 4 12) are inconvenient if many external perturbations are considered simultaneously, since the operators T ( ' ) and C, are field dependent, so equations (1 1) and (1 2) would have to be solved for each perturba- tion. Fortunately, these operators can be eliminated and ESiAp can be expressed directly in terms of U, i.e. without solving perturbation equations involving U. Two methods are used to effect this elimination: the A operator method (introduced by Bartlett and collabora- tors [29]) and the so-called Handy-Schaefer device [30, 311. The A operator method (us d to eliminate T ( ' ) )

tion structure as T ) defined by the linear equation employs the excitation operator A 7 (of the same excita-

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Substituting 6T by T ( ' ) in equation (14) and by A t in equation (1 1) and subtracting the resulting equations one obtains

([e-THeT,T(')]) = (AtlepT(U + [ H , C1 - Cf])eT). (15)

This equation eliminates T ( ' ) and the expression for

( 1 ) - ( 1 ) ( 1 ) Eresp - E + AEresp,

Eresp (1) takes the form

(16) where

and

(18) AE$& = - ( I + Aile-'[H, c,1eT). t The contribution linear in C1 does not appear in equa- tion (18) since it is eliminated using equation (14). The first term in equation (16) is the non-relaxed property whereas the second term is precisely the relaxation con- tribution.

When U is expanded in terms of the orbital substitu- tion operators EL [32],

rs

the first-order property defined by U (i.e. the expectation value of v> can be written as

rs

where Y is the exact (normalized) eigenfunction of H and

VL = (YIELY) (21) is the orbital representation of the exact spin-integrated one-electron reduced density matrix D(r', r) [33]. One can say that the first-order property is a linear form of the matrix elements Us and the coefficients of this form are given by the density matrix '0:. If the density matrix is known then any first-order property can be computed readily from equation (20). Inserting equation (19) into (17) we find

where

DL = (1 +Atle-TE:eT) (23) is the non-relaxed CC density matrix. The matrix 2) is non-symmetric but this does not cause any problems in the evaluation of equation (22). Since the matrix U is symmetric, the antisymmetric part of D does not con-

tribute to the RHS of equation (22). We can thus sub- tract 4 (D - Dt) from V, and obtain an equivalent, symmetric density matrix.

To obtain an expression for the relaxation contribu- tion to the CC density matrix we have to eliminate the CI operator from equation (1 8). This can be done in the following way. Let C be the superoperator such that for any operator X the operator C(X) is the one-electron excitation operator uniquely defined by the CHF-type equation

(6T, I[H, C(X) - C(X)t] + X) = 0. (24) The following formula, valid for arbitrary operators X and Y,

Re(C(Y)IX) = Re(ylc(x)), (25) may be viewed as a particular realization of the Handy- Schaefer device [31]. By Re we denote here the real part of an expression. Since all quantities considered by us are manifestly real, this restriction is not significant. The proof of equation (25) is as follows. First we write the equation defining C(Y) in the complex conjugate form

([H,C(Y) - C(Y)t] + Y)6T,) = 0. (26) Substituting now 6T1 by C(Y) in equation (24) and by C(X) in equation (26) and subtracting the resulting equations one finds, after some manipulation, that

(C(Y)IX) - (YlCW)) + (c(x)c(y)lH) - (~IC(Y)C(X)) = 0. (27)

Taking the real part of equation (27) results immediately in equation (25).

To apply equation (25) to the elimination of the operator C , t from equation (1 8) we rewrite this equation

AE!& = (c1iz), (28 1 in the form

where the one-particle excitation operator Z is defined as

(29)

Z: = -f( 1 + Atle-'[H, Eh]eT), (30)

with the indices i and a running always over the occu- pied and virtual orbitals, respectively. Applying equa- tion (25) gives

= (ulC(Z)), (31)

Inserting equation (19) into (31) we obtain

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Electrostatic interactions between molecules by CCSD 1727

where

(ADresp)h = ( E h W ) ) = 2[c(z)1:1 (33)

and where [C(Z)]t are the amplitudes of the operator C(Z). Equation (33) shows that the relaxation contribu- tion to 2) is obtained by solving the CHF equation (12) with the matrix elements U l replaced by 22:,. As expected, only the particle-hole-type matrix elements of V are modified by the relaxation. The density matrix D(r',r) and the electron density p ( r ) = D(r,r) appearing in equation (4) can be obtained from the expression

DD(r'1 r) = ~ % w ' ) 4 & ) . (34) rs

The working expressions for the CCSD matrix 27; are derived in the next section. The corresponding codes, developed by one of the present authors (T.K.) will be available in the newest version of the Molpro package 1211.

4. Derivation of orbital expressions In the orbital formulation of equations (14), (23), and

(30) in the CCSD approximation we follow the approach of Hampel et al. [34]. The singles and doubles parts of the operator T are expressed as

where EGb = E$E,!'. It is worth noting that the functions Ef!D and E r a are not orthogonal, although they are linearly independent for i # j. Therefore, it is useful to simplify the derivations by the use of the contravariant set of the operators (see [34], [35], and [36], appendix B):

- - E? I = LEU 2 1 1 Eah 11 = L 6 (2Eub V + Eub)l J1 (36)

for the bra operators 6TI, 6T2, and A t . The singles and doubles parts of the At operator will therefore be expanded as,

ia ijah

For an efficient computer implementation of equations (14), (23), and (30) it is essential to express all equations in terms of matrix operations. To achieve this objective we follow the approach of [34] and represent all opera- tors and wavefunctions in terms of matrices and vectors (matrices and vectors will be denoted by bold capital and bold lower case letters, respectively). For the sake of completeness we repeat here the definitions given in [341.

The two-electron integrals with two external orbitals are collected in the so-called internal Coulomb and

exchange matrices, (J"),,, = (kllab) and (K"),, = (akllb), where

(rsltu) = 4r (r)4,?(r) Ir - r'1-I 4, (r ')& (r ')drdr '. s The letters i, j , k, 1, m denote occupied orbitals, a, b, c , d virtual orbitals, and r, s, t , u stand for all orbitals. The integrals with one external orbital are represented by column vectors, (fk), = hok+ Ci[2(akl i i ) - (ailik)] and (k"')), = (aklli). Additionally, we define the matrices L = 2Kki - K'k and vectors Ik" = 2kk" - kiki. The 2

will be represented in form of column vectors zi, ti, and 1', respectively, wher , for example, ( z ' ) ~ = z:. The dou-

and Av, numbered by pairs of occupied orbitals, where (Tij),h = Tfh, etc. Most terms containing two-electron integrals with three external orbitals can be expressed in terms of contractions analogous with the so-called generalized Coulomb and exchange matrices, (J(E"))ah = C,.(abIic)t{ . and = x,.(ailbc)t;., where (ElJ), = Similarly, all terms containing two-electron integrals with four external orbitals can be expressed through the external exchange matrices of the type of ( (Dij))r,7 = C,,DFu(rt us), where

K(Dv)(k) denotes the kth column of the matrix K(Dij).

present work are listed below:

ki

operator, the singles part of the T and A t amplitudes

bles part of T and A 7 will be described by matrices, Tij

D'j = Cij + EU + (Eji7 and C'J = TI] + t'(t') c . The term

Auxiliary quantities introduced in [34] and used in the

X = F + c { 2 J ( E k k ) - K(Ekk) - [(r k (t k t ) k

pk' = Fki + (ti)tfk + X[(t ' ) t l lki + tr(C"L")], (41) I

where ( F ) a h =fa,, and Fki =hi are virtual-virtual and occupied-occupied parts of the Foch matrix, and K $ = ( i k p l ) is the all-occupied part of the exchange integral.

4.1. Matrix expressions for A amplitudes In order to find the A t operator we should solve

equation (14). This linear equation for A t can be rewritten in the form,

(AtI[e-'He', 6T]) = -(eCTHeT6T). (42)

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The RHS of equation (42) is equal to

(e-*HeTEY) = 2(ri)a and (e-THeTE$) = 2(Li')&

(43)

for single and double excitations, respectively. Since the dimension of linear equation (42) quickly

becomes prohibitively large, it should be solved by some iterative method. Therefore, we need the matrix- vector multiplication procedure for the LHS of this equation. Denoting the LHS of equation (42) by

( v ~ ) ~ = (Afl[eCTHeT, E i ] ) and

(Vk'),d = (Atl[eCTHeTl Eit])l (44)

we may write the following expressions for the vk vectors and the Vk' matrices:

i j

(45) i

and

where

- W" = 2Wij - Wji, for an arbitrary matrix Wij; (49)

i

+ ; x[Lk ' (2Ti j - 1

(56) klj l t CjI) - 1 (t ) 1 ;

Eijkl = tr(AkICji); (60)

(61) (M~'),, = C(t'),[&,(idlac) + A!h(iClad)]. ahi

The matrix Hij is defined analogously to the matrix Eij with the singles t amplitude replaced by the 12 amplitude, i.e. HYc = 67i12<. It should be noted that the intermediates YI;"' and Zy are slightly modified quantities Ykj and ZkJ defined in [34], equations (35) and (36).

4.2. Matrix expressions for the non-relaxed density Once the A amplitudes are known, the correlated part

of the non-relaxed CCSD one-electron density can be calculated from the following equations:

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Electrostatic interactions between molecules by CCSD 1729

where DL, (Z3)ac = D:, (a'): = 2704, and (d)! = @ stand for the hole-hole, particle-particle, hole-particle, and particle-hole parts of the density matrix, respectively.

For testing urposes we also implemented the non- relaxed part (3 = 0) of equation (1 1). The non-relaxed

is obtained readily using the code developed for the RHS EOM- CCSD equations [37], implemented recently into the Molpro package [21, 381. Once the T!Jn-re, is available, we can calculate the value of E ( ' ) directly from the non- relaxed part of equation (10). It should be noted that the T ( ' ) operator will be necessary for the calculation of the second-order properties, like the second-order induction energy.

part of the T ( ' ) operator (denoted by Tnon-rel) (1 )

4.3. Matrix expressions for the Z operator For the relaxation part of the CCSD density matrix

we need the Z operator, defined by equation (30). The orbital form of this expression is given by

k j ijk - c ,,yfj + ~ ~ i ,

i zi = 2f' - 211f' - c . 1 1

i k

+ c ( 2 K i j - Jij)dl - c Fij4 + c Lijkj

- c cjkK(Akj)(i) + c kjklE[ikj

-? ( k j k

+ c { - [ t j ( A k ) t + 1 ~ 1 l k i j +I.$$

+ I$kkikj} + Z I K ( I E ) ( k ' - J(I$)'k']

j 1 i

k j j k l

+ tr x KikAkj) - x KjkAkjti

i k

k

+ C tj tr (7 JikIy) - 2 tk tr (JkiII) j k

where the intermediates are defined by the following equations,

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+ 2 c t'(tm)t (tJ)tAimtk; Im

(74)

(75)

(77)

(82) = tr (AkjT").

In the implementation of the equations above we used the programming devices developed in the suite of Molpro codes, in which the necessity of the full integral transformation can be avoided by using the contraction schemes like those used in the definitions of the quan- tities J(Eg), K\E", and K(DV). For programming the terms xi J(B ')(I , xk J(Iy3)(k) and xj J(If)tJ a Molpro routine for evaluation of the QCISD gradients was utilized [39]. The Mk' matrices were programmed using the transformed three external integrals, available in Molpro for the calculation of the triples contribution in the CCSD(T) method [40].

5. Computational details We performed calculations for several representative

van der Waals complexes: (i) two polar molecules: (HF)2 and (H20)2; (ii) two non-polar molecules: (C0)2 and (N2)2; and (iii) a rare gas atom and a polar molecule: He-H20. These systems can be considered as represen- tative of van der Waals molecules bound by electro- static, electrostatic and dispersion, and pure dispersion forces. The strength of the interaction in these com- plexes varies: the binding energy ranges from

M 30cm-' for He-H20 to M 1500cm-' for the water dimer.

To better gauge the influence of the CCSD correlation effects on the electrostatic interactions we computed the total interaction energies as defined in SAPT. The inter- molecular interaction energy E:tpT was represented by the sum of the first- and second-order polarization and exchange contributions [20],

(2) (2) + Eexch-def + Eexch-disp

where Eixch denotes the first-order exchange energy, Ei:A and Edisp (2) are the induction and dispersion energies, and

(2) (2) (2) Eexch-ind 9 Eexch&f, and Eexch-disp are the second-order exchangeinduction, exchangedeformation, and ex- change-dispersion energies. The contributions ap- pearing on the RHS of equation (83) were evaluated with the many-body techniques developed in [lo, 36, 41461 (see also [20] for a review). The exchangdefor- mation term [47, 481 was computed from the supermo- lecular Hartree-Fock interaction energy using equation (10) of [9]. The computational scheme adopted in this work was the same as in [8, 91, except that was computed from equation (8) or (9).

To get an insight into the accuracy of the SAPT results with the electrostatic energy obtained from our best approximation, equation (8), we performed addi- tional calculations of the interaction energies with the supermolecular coupled cluster method restricted to single and double excitations with a non-iterative inclu- sion of triple excitations, CCSD(T) [49]. Nowadays this method is used frequently to provide accurate, bench- mark results, both for molecular properties and for intermolecular potentials. The interaction energy in the supermolecular calculations was corrected for basis set superposition error (BSSE) using the Boys-Bernardi counterpoise correction [50, 5 I].

The geometries of the water, HF, NZ, and CO dimers were taken from [23, 36, 521. In all cases we considered the structures corresponding to the global minima (GM) on the corresponding potential energy surfaces. For the CO dimer we performed additional calculations for the local minimum (LM) and saddle point (SP) geometries [23]. The water goemetry in the He-H20 complex was the same as in [53]. The distance between the helium and oxygen atoms is equal to 5.96874 au, and the angle between the C2u axis of the water molecule and the &He vector is 103.491'. This geometry has been recently found to be the global minimum geometry for the H e H 2 0 complex [54].

Except for the He-H20 molecule, in both the super- molecular and SAPT calculations we used the medium-

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Electrostatic interactions between molecules by CCSD 1731

polarized basis sets of Sadlej [55]. These bases were care- fully optimized for the first- and second-order electric properties of the molecules, and are expected to give a reliable description of the intermolecular interactions. For water in the He-H20 complex we used a combined basis from [56], table XIII: [7s4p/4s] for the isotropic part of the basis, (2dlflg) and (2pld) for the polariza- tion parts for 0 and H atoms, respectively, and 3s2pldlf for midbond (the exponents for the midbond part are the same as in [56] with the exception of the exponent of the f function, which was set equal to the exponent of the f function of oxygen). For helium we used the [7s5p4d3fl basis from [57], optimized for the He-He interaction. All supermolecular calculations were under- taken with the Molpro code [21]. In the SAPT calcula- tions we employed the SAPT system of codes [58, 591. The CCSD electrostatic energies were calculated using equations (14), (23), and (30) implemented into the Molpro code. The code was tested by the comparison of the dipole moment values calculated from (via the RHS of equation (20)) with the results of the numerical differentiation of the CCSD energies computed in the presence of the dipole field, and with the dipole moments computed with the Aces11 program [60]. The non-relaxed CCSD density was tested by calculating the dipole moment directly from the non-relaxed part of equation (lo), as described in $4. Also we considered

the electrostatic energy computed with the electron den- sities from the quadratic configuration interaction theory restricted to single and double excitations (QCISD) [61]. The QCISD electrostatic energies were obtained from the QCISD relaxed density matrices, implemented recently into the Molpro code by Rauhut [391.

6. Results and discussion The results of the perturbative and CCSD calcula-

tions of the electron correlation corrections to the elec- trostatic energy are reported in table 1. Also presented in this table are the relaxation contributions appearing in the second and third order of the Merlller-Plesset theory and at the CCSD level (the latter defined as the difference between the electrostatic energy expression of equation (4) calculated with the relaxed and nonrelaxed one-electron CCSD density matrices). For comparison, the contribution A:f?t,resp(QCISD) analogous to the dif- ference Ak;:t,resp(CCSD) of equation (7), but appearing in the QCISD approximation, is also given. An inspec- tion of table 1 shows that, except for the CO dimer, the convergence of the relaxed Mder-Plesset expansion for the electrostatic energy restricted to single, double, and quadruple excitation diagrams is satisfactory. This is shown by the small size of the LI~~~~,,,,,(CCSD) term, which represents the fifth- and higher-order diagrams

Table 1. Convergence of the M~rller-Plesset perturbation expansion of the electrostatic energy (in cm-I).

2319.26

104.70

68.39

30.73

2.82

2.76

5.01

15.03

-65.23

28.37

-7.61

- 2 197.44

97.1 1

72.97

22.28

- 1.92

3.77

5.19

9.79

- 82.96

40.91

-9.92

- 90.00

- 34.09

-67.50

30.13

3.37

- 0.08

- 3.44

- 12.72

8.60

-21.30

5.31

-45.98

- 8.63

- 24.00

9.44

14.80

- 8.88

-9.55

-4.90

- 1.95

-2.64

- 1.79

-77.80

10.13

23.58

- 6.23

- 14.66

7.44

6.22

1.05

- 10.69

15.24

- 1.03

-51.07

- 16.20

- 19.17

5.22

-2.32

0.07

-0.14

- 1.89

-4.34

- 0.26

1.18

- 6.40

-0.77

-1.10

0.47

-0.22

0.08

0.03

- 0.27

-0.05

0.01

0.12

a Sum of the fifth- and higher-order intramonomer correlation terms included in E&esp(CCSD), cf. equation (7). Sum of the fifth- and higher-order intramonomer correlation terms included in E$Jt,resp(QCISD), cf. equation (7). Relaxation contribution through the second order, AE(’)

Relaxation contribution at the CCSD level defined as a difference between the eictrostatic energy expression of equation (4)

(2) = (Ifilst,resp E(I2) - (lfjst E(12) . Relaxation contribution through the third order, AEelsi,resp(3) (I)elst,resp = Eelst,resp + Ee,st,res - EL;:) - ELI:).

calculated with relaxed and nonrelaxed CCSD density matrices.

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1732 T. Korona et al.

summed to infinite order in CCSD theory. For the global minimum geometries dLijt,,,,,(CCSD) amounts

Somewhat unexpectedly, the largest value of dLl,),,res,(CCSD) relative to (4) is obtained for the weakly bound He-H20 complex. Since dL;:t,resp(CCSD) for the global minimum of the CO dimer is suspiciously small, we performed additional calculations for the local minimum and the saddle point geometries. As expected, it turned out that the small value of AL~~t,r,,,(CCSD) for the global minimum was accidental. For the LM and SP geometries this term is almost as large as the total correlation contribution to Ek;/t,resp(CCSD) given in the second row of table 1. It is clearly seen that the relaxed Maller-Plesset expansion for the electrostatic interaction energy in the CO dimer cannot be considered as convergent. Except for the CO

appears to be more important than the fifth- and higher-order terms from the CCSD theory. The latter conclusion is especially true for the N2 dimer, which represents an example of the electrostatic interactions between molecules with triple bonds. Surprisingly, it also holds to some extent for the weakly bound van der Waals complex He-H20, and does not apply to typical hydrogen bonded systems like the H 2 0 and HF dimers. For the CO dimer the role of the triples contri- bution varies considerably with the geometry. Since this term changes sign, it can be expected that it goes through zero somewhere close to the SP geometry, and the relatively minor importance of this term for the SP

to M 0.1-1.0% of Eelst,resp ( 1 ) (4), and thus is negligible.

dimer, the triple excitation contribution Ee,st,resp(T) (14)

geometry is also accidental. The relaxation contribution dE:i&,,,(CCSD) in the CCSD approximation is of the same order as the term ELt,",lres,(~) and thus cannot be neglected in any accurate application. It is interesting to observe, cf. the last three rows of table 1, that the term dEL;Jt,,,,,(CCSD) is usually much smaller than the relaxation contributions to the electrostatic energies appearing in the second- and third-order of the Marller-Plesset theor . A comparison of the infinite-

shows that the differences in the correlation contribu- tions predicted by the CCSD and QCISD theories are noticeable. Summarizing, the results presented in table 1 show that it may be worth considering the effect of the converged CCSD amplitudes on the triple excitation contribution to

Table 2 reports the SAPT results for the total inter- action energy and its components. To better appreciate the importance of high-order correlation effects, the electrostatic term was computed either perturbatively through the fourth order in the intramonomer correla- tion, cf. equation (9), or with the inclusion of the Mth- and higher-order terms from the CCSD theory, cf. equation (8). As might be expected, the inclusion of the dL{:t,,,sp(CCSD) term does not lead to significant changes in the total interaction energies with the excep- tion of the CO dimer, where it even changes the ener- getic order of the LM and SP geometries. The SAPT

fers marginally from EZtpT that includes in addition the CCSD correction LIL;~~,,~,,(CCSD). For the hydrogen

order corrections d,,,,,,,(CCSD) (IT and d,,,,,,,,,(QCISD) ( 1 )

in the spirit of CCSD(T) theory.

interaction energy with the Eelst,resp (1) (4) contribution dif-

Table 2. Components of the interaction energy (in cm-') calculated for various complexes.

-2202.29

- 2 199.53

1927.1 1

-782.46

388.88

- 703.08

93.71

- 199.62

- 1477.75

- 1474.99

- 1480.89

- 2094.32

- 2090.55

2055.1 1

- 1007.59

475.42

-600.19

70.82

- 209.42

-1310.17

- 1306.40

- 1349.86

- 136.73

-136.81

225.33

-44.65

36.85

-201.45

11.75

- 17.27

-126.17

- 126.25

- 85.95

- 50.63

-59.51

135.79

- 24.48

21 s o -181.59

9.06

- 5.98

-96.33

- 105.21

-76.95

-74.08

-66.64

122.73

-26.18

23.04

- 154.70

8.05

- 6.47

- 107.60

- 100.16

- 87.52

- 69.22

-69.16

1 16.95

- 20.07

17.81 - 143.76

6.64

- 5.09

-96.75

- 96.68

- 87.90

- 7.52

- 7.44

39.38

-6.85

1.75 -62.13

2.06

-2.01

-35.32

-35.24

-34.17

'The electrostatic term was approximated by EL:,)l.rcsp (4), see equation (9). The electrostatic term was approximated by @jt,resp, see equation (8).

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Electrostatic interactions between molecules by CCSD 1733

bonded dimers ((H20)2 and (HF)2) the difference is smaller than 0.3%, and for the nitrogen dimer and the He-water complex it is even smaller, 0.07% and 0.2%, respectively. For the CO dimer the difference depends strongly on the geometry. For the global minimum it is small and represents 0.06% of the interaction energy, while for the local minimum and the saddle point geo- metries the difference between the two SAPT results is appreciable and amounts to 9% and 7% for the LM and SP geometries, respectively. It is surprising to note that for this dimer the CCSD(T) interaction energy of GM is slightly higher than the energy of SP. However, the order of the MP4 supermolecular energies is correct: the MP4 interaction energies are equal to - 1 17.3 1, - 1 12.12, and -83.00cm-', for the GM, LM, and SP geometries, re- spectively (these geometries were taken from [23], where the potential energy surface was investigated by means of the supermolecular MP4 calculations).

It was shown in [62] that for simple closed shell dimers the agreement between the SAPT and supermolecular CCSD(T) calculations is usually very good, the differences being of the order of a few per cent. A comparison of the SAPT and CCSD(T) results reported in table 2 supports these conclusions. Table 2 shows that for simple complexes, like the water and HF dimers and the He-H20 complex, the two sets of results agree very well. Somewhat surprisingly, the inclusion of LI~,!~~,~,,,(CCSD) slightly worsens this agreement for (H20)2 and (HF),. The situation is quite different for the carbon monoxide and nitrogen dimers. Here the differences between the best SAPT results and the CCSD(T) data are appreciable, even if higher order intramonomer correlation is accounted for in the electrostatic term. In order to find the reason for such a disagreement, we computed for the CO dimer in its global minimum geometry the third-order intra- monomer correlation correction to the dispersion energy resulting from the Mraller-Plesset expansion of the CCD + ST(CCD) dispersion energy as defined in [63]. It amounts to 42.55 cm-' [64] and, if added to the SAPT interaction energy, gives a result in very good agreement with CCSD(T). Thus, more refined models of the dis- persion interactions are necessary to quantitatively describe the van der Waals interactions between mol- ecules with triple bonds. One may note that such good agreement between SAPT and CCSD(T) as described above may to some extent be fortuitous, since the CCSD(T) theory is not expected to be very accurate in the case of the CO dimer [23, 641.

7. Conclusion The results reported in this paper show that except for

the CO dimer the convergence of the relaxed Mraller- Plesset expansion for the electrostatic energy, restricted

to single, double, and quadruple excitation diagrams, is satisfactory. The role of the fifth- and higher-order terms included in the CCSD theory is very small, except for the pathological case of (C0)2. The triple excitation contribution at the fourth-order Mraller- Plesset level is relatively more important. Comparison of the SAPT and CCSD(T) results shows good agree- ment in the case of the (H20)2, (HF)2, and He-H20 dimers. Some disagreement observed for the cases of the carbon oxide and nitrogen dimers is caused most likely by higher-order intramonomer correlation correc- tions to the dispersion energy not available in the cur- rent implementation of the SAPT program. It appears that a more elaborate model for the dispersion inter- action would be needed to obtain a better agreement with CCSD(T) results.

This work was supported by the KBN through the University of Warsaw (Grant BW-1453/6/99).

References [l] BUCKINGHAM, A. D., FOWLER, P. W., and HUTSON, J.

[2] BUCKINGHAM, A. D., FOWLER, P. W., and STONE, A J.,

[3] DYKSTRA, C. E., 1988, Accounts chem. Res., 21, 355. [4] BUCKINGHAM, A. D., 1967, Adv. chem. Phys., 12, 107. [5] STONE, A. J., 1981, Chem. Phys. Lett., 83, 233. [6] STONE, A. J., and ALDERTON, M., 1985, MoIec. Phys., 56,

1047. [7] SOKALSKI, W. A., and POIRIER, R. A,, 1983, Chem. Phys.

Lett., 98, 86; SOKALSKI, W. A., and SAWARYN, A., 1987, J. chem. Phys., 87, 526.

[8] MOSZYNSKI, R., KORONA, T., WORMER, P. E. S., and VAN DERAVOIRD, A., 1997, J. phys. Chem. A, 101,4690.

[9] HEIJMEN, T. G. A., KORONA, T., MOSZYNSKI, R., WORMER, P. E. S., and VAN DER AVOIRD, A,, 1997, J. chem. Phys., 107, 902.

[lo] MOSZYNSKI, R., JEZIORSKI, B., RATKIEWICZ, A., and RYBAK, S., 1993, J. chem. Phys., 99, 8856.

[Il l SADLEJ, A. J., 1981, J. chern. Phys., 75, 320; DIERCKSEN, G. H. F., and SADLEJ, A. J., 1981, J. chem. Phys., 75, 1253.

[12] SALTER, E. A., TRUCKS, G. W., FITZGERALD, G., and BARTLETT, R. J., 1987, Chem. Phys. Lett., 141, 61.

[13] MOSZYNSKI, R., WORMER, P. E. S., and VAN DER AVOIRD, A., 2000, Computational Molecular Spectroscopy, edited by P. Jensen and P. Bunker (New York: Wiley), p. 69.

[14] LARSEN, H., HALKIER, A., OLSEN, J., and JPIRGENSEN, P., 2000, J. chem. Phys., 112, 1107.

[15] OLSEN, J., JBRGENSEN, P., HELGAKER, T., and CHRISTIANSEN, O., 2000, J. chem. Phys., 112,9736.

[16] BARTLETT, R. J., 1995, Modern Electronic Structure Theory, Part 2, edited by D. R. Yarkony (Singapore: World Scientific) p. 1047.

[17] SCHEINER, A. C., SCUSERIA, G. E., RICE, J. E., LEE, T. J., and SCHAEFER, H. F., 1987, J. chem. Phys., 87, 5361.

[18] SALTER, E. A,, TRUCKS, G. W., and BARTLETT, R. J., 1989, J. chem. Phys., 90, 1752.

M., 1988, Chem. Rev., 88, 963.

1986, Zntl Rev. phys. Chem., 5, 107.

Dow

nloa

ded

by [

Penn

sylv

ania

Sta

te U

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] at

07:

48 3

0 Ju

ne 2

012

1734 Electrostatic interactions between molecules by CCSD

[19] HALKIER, A,, KOCH, H., CHRISTIANSEN, o., J0RGENSEN, P., and HELGAKER, T., 1997, J. chem. Phys., 107, 849.

[20] JEZIORSKI, B., MOSZYNSKI, R., and SZALEWICZ, K., 1994, Chem. Rev., 94, 1887.

[21] WERNER, H.-J., KNOWLES, P. J., AMOS, R. D., BERNHARDSSON, A., BERNING, A., CELANI, P., COOPER, D. L., DEEGAN, M. J. O., DOBBYN, A. J., ECKERT, F., HAMPEL, C., HETZER, G., KORONA, T., LINDH, R., LLOYD, A. W., MCNICHOLAS, S. J., MANBY, F. R., MEYER, W., MURA, M. E., NICKLASS, A., PALMIER], P., PITZER, R., RAUHUT, G., SCHUTZ, M., SCHUMANN, U., STOLL, H., STONE, A. J., TARRONI, R., and THORSTEINSSON, T., 1998, Molpro, a package of ab initio programs.

[22] ELAZHARY, A,, RAUHUT, G., PULAY, P., and WERNER, H.-J., 1998, J. chem. Phys., 108, 5185.

[23] RODE, M., SADLEJ, J., MOSZYNSKI, R., WORMER, P. E. S., and VAN DER AVOIRD, A., 1999, Chem. Phys. Lett., 314, 326.

[24] KUCHARSKI, S. A., and BARTLETT, R. J., 1998, J. chem. Phys., 108, 9221.

[25] RODE, M., SADLEJ, J., MOSZYNSKI, R., WORMER, P. E. S., and VAN DER AVOIRD, A., 2001, Chem. Phys. Lett., 334, 424.

[26] PEDERSEN, T. B., FERN~NDEZ, B., and KOCH, H., 2000, Chem. Phys. Lett., 334, 419.

[27] BUKOWSKI, R., JEZIORSKI, B., and SZALEWICZ, K., 1998, J. chem. Phys., 108, 7946.

[28] MONKHORST, H. J., 1977, Intl J. Quantum Chem. S, 11, 421; DAALGARD, E., and MONKHORST, H. J., 1983, Phys. Rev. A, 28, 1217.

[29] SALTER, E. A., TRUCKS, G. W., and BARTLETT, R. J., 1989, J. chem. Phys., 90, 1752.

[30] PULAY, P., 1995, Modern Electronic Structure Theory, Part 2, edited by D. R. Yarkony (Singapore: World Scientific), p. 1191.

[31] HANDY, N. C., and SCHAEFER, H. F., 1984, J . chem. Phys., 81, 5031.

[32] PALDUS, J., 1976, Theoreticaf Chemistry Advances and Perspectives, Vol. 2, edited by H. Eyring and D. Henderson (New York: Academic Press), p. 177.

[33] DAVIDSON, E. R., 1976, Reduced Density Matrices in Quantum Chemistry (New York: Academic Press).

[34] HAMPEL, C., PETERSON, K., and WERNER, H.-J., 1992, Chem. Phys. Lett., 190, 1.

[35] PULAY, P., SEB0, S., and MEYER, W., 1984, J. chem. Phys., 81, 1901.

[36] RYBAK, S., JEZIORSKI, B., and SZALEWICZ, K., 1991, J. chem. Phys., 95, 6576.

[37] COMEAU, D. C., and BARTLETT, R. J., 1993, Chem. Phys. Lett., 207, 414; STANTON, J. F., and BARTLETT, R. J., 1993, J. chem. Phys., 98, 7029; GWALTNEY, S. R., NOOIJEN, M., and BARTLETT, R. J., 1996, Chem. Phys. Lett., 248, 189.

[38] KORONA, T., and WERNER, H.-J., unpublished. [39] RAUHUT, G., personal communication. [40] DEEGAN, M. J. O., and KNOWLES, P. J., 1994, Chem.

[41] SZALEWICZ, K., and JEZIORSKI, B., 1979, Mofec. Phys., Phys. Lett., 227, 321.

38, 191.

[42] JEZIORSKI, B., MOSZYNSKI, R., RYBAK, S., and SZALEWICZ, K., 1989, Many-Body Methods in Quantum Chemistry, edited by U. Kaldor (New York: Springer- Verlag), p. 65.

[43] MOSZYNSKI, R., JEZIORSKI, B., and SZALEWICZ, K., 1994, J. chem. Phys., 100, 1312.

[44] MOSZYNSKI, R., JEZIORSKI, B., RYBAK, S., SZALEWICZ, K., and WILLIAMS, H. L., 1994, J. chem. Phys., 100, 5080.

[45] MOSZYNSKI, R., CYBULSKI, S. M., and CHALASINSKI, G., 1994, J . chem. Phys., 100,4998.

[46] MOSZYNSKI, R., JEZIORSKI, B., and SZALEWICZ, K., 1993, Intl J. Quantum Chem., 45, 409.

[47] JEZIORSKA, M., JEZIORSKI, B., and CIZEK, J., 1987, Intl J. Quantum Chem., 32, 149.

[48] MOSZYNSKI, R., HEIJMEN, T. G. A., and JEZIORSKI, B., 1996, Molec. Phys., 88, 741.

[49] RAGHAVACHARI, K., TRUCKS, G. W., POPLE, J. A., and HEAD-GORDON, M., 1989, Chem. Phys. Lett., 157, 479.

[50] BOYS, S. F., and BERNARD], F., 1970, Molec. Phys., 19, 553.

[51] VAN DUIJNEVELDT, F. B., VAN DUIJNEVELDT-VAN DE RIJDT, J. G. C. M., and VAN LENTHE, J. H., 1994, Chem. Rev., 94, 1873.

[52] WADA, A., KANAMORI, H., and IWATA, S., 1998, J. chem. Phys., 109, 9434.

[53] MAS, E. M., and SZALEWICZ, K., 1996, J. chem. Phys., 104, 7606.

[54] PATKOWSKI, K., MOSZYNSKI, R., KORONA, T., and SZALEWICZ, K., unpublished.

[55] SADLEJ, A. J., 1988, Collect. Czech. Chem. Cornmun., 53, 1995.

[56] WILLIAMS, H. L., MAS, E. M., SZALEWICZ, K., and JEZIORSKI, B., 1995, J. chem. Phys., 103, 7374.

[57] KORONA, T., WILLIAMS, H. L., BUKOWSKI, R., JEZIORSKI, B., and SZALEWICZ, K., 1997, J. chem. Phys., 106, 5109.

[58] JEZIORSKI, B., MOSZYNSKI, R., RATKIEWICZ, A., RYBAK, S., SZALEWICZ, K., and WILLIAMS, H. L., 1993, Methods and Techniques in Computational Chemistry, METECC-94, Vol. B, Medium Size Systems, edited by E. Clementi (Cagliari: STEF) p. 79.

[59] BUKOWSKI, R., JANKOWSKI, P., JEZIORSKI, B., JEZIORSKA, M., KUCHARSKI, S. A., MOSZYNSKI, R., RYBAK, S., WILLIAMS, H. L., and WORMER, P. E. S., 1996, SAPT96: An ab initio program for many-body sym- metry-adapted perturbation theory calculations of inter- molecular interaction energies (University of Delaware and University of Warsaw).

[60] STANTON, J. F., GAUSS, J., WATTS, J. D., LAUDERDALE, W. J., and BARTLETT, R. J., 1992, Aces I1 Program System; Zntl J. Quantum. Chem., S26, 879.

[61] HEAD-GORDON, M., POPLE, J. A., and RAGHAVACHARI, K., 1987, J. chem. Phys., 87, 5968.

[62] MILET, A., KORONA, T., MOSZYNSKI, R., and KOCHANSKI, E., 1999, J. chem. Phys., 111, 7727.

[63] WILLIAMS, H. L., SZALEWICZ, K., MOSZYNSKI, R., and JEZIORSKI, B., 1995, J. chem. Phys., 103, 8058.

[64] MILET, A., KORONA, T., KUCHARSKI, S. A., and MOSZYNSKI, R., unpublished.

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