Absolutely Local Excited Orbitals in the Higher Order Perturbation

Sep 25, 2008 - It is demonstrated that the third order calculation is a practical and ... local excited orbitals is a compromizing technique to take i...
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16104

J. Phys. Chem. B 2008, 112, 16104–16109

Absolutely Local Excited Orbitals in the Higher Order Perturbation Expansion for the Molecular Interaction† Suehiro Iwata* Center of Quantum Life Sciences and Graduate School of Science, Hiroshima UniVerity, Higashi-hiroshima, 739-8526 Japan, and Toyota Physical and Chemical Research Institute, Nagakute, Aichi, 480-1192 Japan ReceiVed: July 3, 2008; ReVised Manuscript ReceiVed: August 7, 2008

Based on the locally projected molecular orbitals, the third and fourth order perturbation corrections for the molecular interaction within the single excitations are evaluated, and the calculated interaction energies are compared with the counterpoise (CP) corrected interaction energy of the Hartree-Fock level of theory. It is demonstrated that the third order calculation is a practical and powerful method to obtain the binding energy almost equal to the CP corrected energy. It requires only one more two-electron integral handling after the LP MO calculation, which is faster than a usual supermolecule HF calculation. For the perturbation expansion, the absolutely local excited orbitals are determined. For small basis sets, it is shown that the partial delocalization of the absolutely local excited orbitals is a compromizing technique to take into account the charge-transfer contribution without reintroducing a large basis set superposition error. Introduction In the past two decades, great progress has been made in theories and experiments on the molecular interaction and molecular clusters, and many review articles and text books have been published.1-4 In recent years, computational studies have become indispensable tools for experimentalists to analyze their molecular spectra and mass spectra of molecular clusters. Now most of the molecular spectroscopists carry out the quantum chemical calculations by themselves to identify the species and states they detected. The quantum chemical calculations look like a part of the experimental instruments. However, the outputs of the computations are not straightly the scientifically valuable information, just like those of the experimental instruments are not. In particular, in calculations involving the weak molecular interaction, special care is required to adopt a well-balanced approximation for the interacting system and the isolated constituents. To properly evaluate the molecular interaction energy, it is essential to have consistency in the approximation methods for the molecular systems involved. As Liu and McLean discussed,5 there are two aspects of the consistency in the many-electron theory for the molecular interaction; one is the orbital basis inconsistency (OBI) and the other is the configuration basis inconsistency (CBI). They are interrelated with each other, but they have to be distinguished. The OBI results from the incompleteness of the one-electron basis functions. The error caused by the OBI is called the basis set superposition error (BSSE). The counterpoise (CP) procedure is intended to remove the OBI, but it is not clear in what condition the CP procedure is able to remove the inconsistency for the many-electron configuration functions (CBI).6 For the Hartree-Fock theory of a single electronic configuration, the wave function is variationally determined with a given basis set at every step of the CP procedure, and the CP correction removes the OBI. On the other hand, for the multiconfiguration † Part of the “Karl Freed Festschrift”. * To whom correspondence should be addressed. E-mail: riken-iwata@ mosk.tytlabs.co.jp.

theory such as multiconfiguration self-consistent field (MCSCF), as was carefully analyzed by Liu and McLean for the He dimer,5 the choice of the molecular orbitals to construct the electron configurations does influence the calculated binding energy. The CP procedure is now routinely used for the MP2 (Møller-Plesset second order) approximation using the full set of the supermolecule’s molecular orbitals (MOs), but it is not well analyzed in the author’s knowledge whether the CP procedure using the full MO space at each step removes the CBI with a finite oneelectron basis set, although it is known that the CP correction for the MP2 energy for a series of basis sets of aug-cc-pVxZ (x ) D, T, Q, 5, 6) converges very slowly to the complete basis set (CBS) limit.7 Mata and Werner demonstrated that, by using the restricted virtual space projected on to the atomic orbitals proposed by Pulay,8 the BSSE can be reduced and better basis set convergence is obtained.9 This work clearly shows that an efficient removal of the CBI in the correlated level of theories requires careful restrictions on the excited orbital space and on the many-electron configuration space. To avoid, to reduce, or to remove the OBI and CBI, there are several strategies other than the CP procedure. The most straightforward way applicable to both OBI and CBI is the extrapolation to CBS after a series of the calculations with augcc-pVxZ or its approximation by assuming that the correlated correction evaluated with a smaller basis set can be trasferable to the CBS limit.10 The method is suitable for the benchmark calculations and not practical for the large size of clusters. The symmetry adapted perturbation (SAPT) is an alternative method and is recently being used,3,11 but the target system is mostly the dimer. The other strategy suitable to the clusters of many constituent molecules (atoms) is to break down the molecular interaction to the sum of the binary and/or ternary interactions, which are evaluated by calculating the dimers and trimers in the cluster. Under this strategy, Kitaura and co-workers developed the method called fragment molecular orbitals (FMO).12 Mochizuki and co-workers combine the FMO with the local MP2.13 Hirata and co-workers generalize the strategy to be applicable to any correlated levels of the theory together with

10.1021/jp805883c CCC: $40.75  2008 American Chemical Society Published on Web 09/25/2008

Absolutely Local Excited Orbitals the integral application.14 Although these methods can avoid the time-consuming counterpoise correction procedure for the large size of the cluster, the correction terms for every dimer and trimer in the cluster should be evaluated, and therefore, the CBI has to be carefully examined for the multiconfiguration wave functions. The use of the orbitals localized on each constituent is another alternative strategy. The local MP2 of Pulay is one of them,8 but because the occupied orbitals are the localized orbitals transformed from the canonical orbitals, the BSSE in the Hartree-Fock level has to be removed for the molecular interaction. For the different context, Stoll first derived the Hartree-Fock equation under the restriction15 that the occupied MOs are expanded in terms of the basis sets only on some of atoms in a molecule. Gianinnetti and his collaborators developed the self-consistent field MO for molecular interaction (SCF MO MI) to evaluate the weak molecular interaction energy without the BSSE.16,17 Later Nagata and Iwata reformulated the set of equations for SCF MO MI, using the projection operator, and called the determined MO locally projected MO (LPMO).18 They analytically proved that the Slater determinant wave function constructed from LPMO cannot include the chargetransfer (CT, electron-delocalization) between the molecules, however large the basis set for each molecule is. Because of the lack of the CT, the evaluated binding energy is systematically underestimated even at aug-cc-pVQZ.18,19 Khaliullin and HeadGordon called these MOs the absolutely local MO (ALMO),20 and they used them to analyze the wave function.21 To introduce the electron-delocalization in a controlled way in the LPMO (ALMO) based wave function, Iwata defined the locally projected excited orbitals for the perturbation expansion.22,23 Because the LPMOs are not the canonical MOs for the Fock operator of the supermolecule, the single excitations from the reference Slater determinant contribute to the second order perturbation correction E(2). He demonstrated that the binding energy evaluated with the second order single-excitation perturbation expansion E(2)SPT becomes close to those of the CP corrected HF, particularly if the locally projected excited MOs are properly selected in the calculations. However, even with the aug-cc-pVQZ basis set a small difference persists in E(2)SPT; the error is usually less than 1 kJ/mol in most of the examples. The purpose of the present paper is to remove this systematic error.22 The present work shows that this error can be removed by the third E(3)SPT and fourth E(4)SPT order corrections. It is demonstrated that the third order calculation is a practical and powerful way to obtain the binding energy almost equal to the CP corrected energy; it requires only one more two-electron AO integral handling after the LPMO calculation, which is faster than a usual supermolecule HF calculation. Theoretical The locally projected self-consistent field molecular orbital method is already described in the literature.16-18,20-24 The method is developed to study the molecular clusters consisting of weakly interacting molecules. The occupied molecular orbitals are locally expanded in terms of the basis sets defined on each molecule which constitutes the cluster. The molecular orbital coefficients for the occupied orbitals are variationally determined under this restriction. Because of this restriction, the calculated interaction energy is free of the basis set superposition error, but because the wave function is absolutely local, it fails to describe the electron delocalization (chargetransfer, CT) among the molecules. It is analytically proved18

J. Phys. Chem. B, Vol. 112, No. 50, 2008 16105 that with the Mulliken population analysis the number of electrons in each molecule is fixed at that of the free molecule. The lack of the electron delocalization (CT) causes the large underestimation for the binding energy,18,20,22,23 and therefore, the perturbation correction to the energy and wave function is required to incorporate the charge-transfer terms. However, the charge-transfer electron configurations in the perturbation expansion possibly reinduce the BSSE (the details will be demonstrated later). To control the contribution from the chargetransfer terms, the well-defined local excited (virtual) orbitals are required in the expansion. To extend the locally projected nature of the occupied MOs to the excited MOs, the eigenvalue problem for a column vector ex of the MO coefficients for φ tAk Ak ex (1 - ˆ Pocc)χAtex Ak ) χAtAkηk

(1)

is solved, where Pˆocc is a projection operator for the space spanned by all of the occupied MOs of the cluster and χA is a row vector of the basis functions defined only on molecule A. For each molecule in the cluster, a similar equation is set up. If ex is orthogonal to all of the occupied ηk ) 1, MO φAk ≡ χAtAk MOs of the cluster, even with the restriction on the basis sets. The orthogonality to the occupied MOs is required for the perturbation expansion. We may call these MOs the absolutely local excited MO (ALExMO, or strictly monomer basis excited MO). In the present study, when ηk g 0.99999, the orbitals are classified as ALExMO. For ηl < 0.99999, the excited MOs are partially delocalized as ex φAl ) NAl(1 - ˆ Pocc)χAtAl

(2)

ex is where NAl is a normalization factor. Because the vector tAl an eigenvector of (1), the partially delocalized excited MO is also characterized by the eigenvalue ηl. In the computer codes, the excited MOs for each molecule are Schmidt-orthogonalized in the decreasing order of η. Although the MOs are not the canonical orbitals for the full ˆ 0 is defined similar Fock operator, the zero order Hamiltonian H 20,22,23 to the Møller-Plesset form as

ˆ0) H

occ

exct

b,c

r,s

∑ ˆa†b〈b|Fˆ|c 〉 ˆac + ∑ ˆa†τ〈r|Fˆ|s〉aˆs

(3)

and therefore the perturbation term is split to the one- and twoelectron parts as

ˆ)H ˆ 0 + λ(V ˆ +ˆ H V2) 1 ˆ + λV ˆ )F 2

(4) (5)

occ exct

ˆ ) λV 1

∑ ∑ ˆa†b〈b|Fˆ|s〉aˆs + c.c. b

(6)

s

In the previous works,20,22,23 only term λVˆ1 is taken into account within the single excitation perturbation (SPT). In the present work, the contribution from term λVˆ2 is examined, which starts to appear at the third and fourth order perturbation terms. As is well-known, the third order energy E(3) can be evaluated by the (1) as first order wave function aa, r occ exct occ exct

E(3) )

∑ ∑ ∑ ∑ ab,t(1)〈(b f t)s|λVˆ2|(a f r)s〉aa,r(1) b

s

(7)

τ

a

occ exct occ exct

)

∑ ∑ ∑ ∑ ab,t(1){2(a_r|tb_) - (a__b|tr)}aa,r(1) b

t

a

(8)

r

where (a f r)S stands for the singlet excited configuration function, and the underbars in MO integrals (ar|tb) imply that

16106 J. Phys. Chem. B, Vol. 112, No. 50, 2008

Iwata

the occupied MOs in the integrals are biorthogonal transformed among the occupied orbitals.19 The coefficients a(1) a, r are a solution of a set of linear equations.19 To evaluate E(3), the MO integral transformation is not required; by defining the first order density matrix correction Γ(1) in terms of the basis sets (u, V) occ exct

(1) Γu,V )

∑ ∑ aa,r(1) tua_ tVr a

the third order correction energy is basis_sets

E(3) )

∑ w,x

[∑

basis_sets

(1) Γw,x

u,V

(9)

r

]

(1) {2(uV|xw) - (uw|Vx)}Γu,V (10)

basis_sets

≡

∑

(1) Γw,x Gx,w

(11)

w,x

and matrix G can be obtained by calling a subroutine used in constructing the Fock matrix. As was described in the previous papers,22,23 the required CPU cost for obtaining the LP SCF MO is less than that for a single supermolecule calculation. Besides, because no handling of the two-electron integrals is required for the second order correction, E(2) can be quickly evaluated. To evaluate E(3), the cost is less than that of a single SCF cycle. To examine the convergence of the perturbation expansion, E(4) is also calculated, and it requires the second order wave (2) , which is a solution of a set of linear equations function aa, r (1) . However, the inhomogeneous terms similar to the one for aa, r require the two-electron MO integrals. The perturbation correction energy E(n) is a function of η, the eigenvalue of (1); E(n)(0.99999) is the correction energy with the use of the absolutely local excited (ALEx) MOs and E(n)(0) is the energy evaluated with all of the excited orbital. Equation 5 implies that the diagonalization of Fock operator Fˆ ˆ 0 + λVˆ1 gives the full correction energy EV1 from λVˆ1, as )H was mentioned in ref 20, and it is also a function of η, EV1(η). In most of our test calculations shown in the next section, E(2)(η) is nearly equal to EV1(η), so that only E(2)(η) is shown. The present version of codes was developed using the Intel compiler on Intel Mac and on Linux as one of the tasks of the MOLYX package, which is a private ab initio MO package and uses the integral package of the 1997 version of GAMESS.25 The counterpoise corrections are evaluated with Gaussian 0326 partly at RCCS, Okazaki Research Facilities, National Institutes of Natural Sciences. Computational Results The single excitations from the LP SCF configuration are grouped into the charge-transfer (CT) and locally excited (LE) configurations. In the following calculations, both configurations are included. Test calculations excluding the LE are carried out. Also test calculations removing the excitations from the core orbitals are examined. Because the changes in the energy are not significant in the following discussion, all of the calculational results are those of all single excitations with the core orbitals. Figure 1 shows the potential energy curves of HF dimers calculated with (a) aug-cc-pVQZ, (b) aug-cc-pVDZ, and (c) ccpVDZ basis sets. The geometric parameters are optimized with MP2/aug-cc-pVDZ for a given F-F distance. Note that the scale of the ordinate is different for the three panels. Figure 1a) clearly demonstrates that the curves of the third E(3) and fourth order E(4) corrections with aug-cc-pVQZ are extremely close to the counterpoise (CP) corrected curve; the difference is less than 0.1 kJ/mol over the examined bond lengths. The BSSE with

Figure 1. Interaction potential energy curves of the HF dimer. (a) The basis set used is aug-cc-pVQZ, (b) the basis set is aug-cc-pVDZ, and (c) the basis set is cc-pVDZ. Filled circle: uncorrected SCF. Filled square: Counterpoise corrected SCF. Filled leftward triangle: LP SCF. Filled rhombus: E(2)(0.99999). Unfilled rhombus: E(2)(0). Filled downward triangle: E(3)(0.99999). Unfilled downward triangle: E(3)(0). Filled upward triangle: E(4)(0.99999). Unfilled upward triangle: E(4)(0).

this large basis set is small (about 0.2 kJ/mol), as is seen in the figure. Yet the closeness of three curves (third, fourth SPT and CP corrected) is remarkable. With this basis set the use of all excited MOs does not change the curves from the restricted use of the absolutely local excited (ALEx) MOs; the filled and unfilled triangle points are overlapped in the curves. On the other hand, for smaller basis sets, this difference becomes clear. In Figure 1, panels b and c, the curves of the unfilled triangle points for the third E(3)(0) and fourth order

Absolutely Local Excited Orbitals

Figure 2. Basis set dependence of the binding energy. (a) HF dimer at RF-F ) 2.8, (b) H2O dimer at RO-O ) 2.9. Filled circle: uncorrected SCF. Filled square: Counterpoise corrected SCF. Filled leftward triangle: LP SCF. Filled rhombus: E(2)(0.99999). Unfilled rhombus: E(2)(0). Filled downward triangle: E(3)(0.99999). Unfilled downward triangle: E(3)(0). Filled upward triangle: E(4)(0.99999). Unfilled upward triangle: E(4)(0).

E(4)(0), which are evaluated by using all of the excited orbitals, overshoots the CP corrected curves. With aug-cc-pVDZ, they are only slightly lower than the CP corrected curves, but with cc-pVDZ, the curve E(4)(0) (unfilled upward triangle) almost coincides with the uncorrected SCF curve (filled circle); that is, the BSSE is recovered by the fourth order correction, if all of the excited orbitals are used in the perturbation expansion. If only the absolutely local excited orbitals (ALEx) are used for aug-cc-VDZ, the curves of E(3)(0.99999) and E(4)(0.99999) are slightly higher than that of the CP corrected (the curve of E(3)(0.99999) is accidentally overlapped with the curve of E(2)(0) (unfilled rhombus) and cannot be seen in the figure). The CP corrected curve is bracketed by E(3)(0.99999) and E(3)(0) (note that this characteristics is basis-set dependent as is seen in Figure 2). The substantial gain is obtained by the third order correction for aug-cc-pVDZ, independent of the threshold value η used, but for cc-pVDZ, if only the absolutely local excited MOs are used (η ) 0.99999), almost no gains are obtained by the third and fourth order correction, as is seen in Figure 1c. With this small basis set, if all of the excited MOs are included in the calculations, even the second order correction E(2)(0) overestimates the binding energy. This is a clear demonstration that the use of the partially delocalized MO introduces the BSSE and that the stabilization by the charge-transfer interaction and the error (BSSE) caused by the orbital basis inconsistency (OBI)

J. Phys. Chem. B, Vol. 112, No. 50, 2008 16107 cannot be separated. So if all of the excited orbitals are used in E(n) and EV1, a kind of counterpoise correction become necessary as is done in ref 21. As is seen in Figure 1, independent of the size of the basis sets and of the threshold value η, the change from E(3) to E(4) is very small, although the required computer costs for E(4) are much larger than those for E(3). Within the single excitation perturbation theory (SPT), the third order correction is nearly converged. For water dimers, the basis set dependence on the potential energy curves is very similar to Figure 1a-c. Figure 2 shows the basis set dependence of the binding energy for HF dimer (Figure 2a) and for water dimer (Figure 2b). The binding energy of the LP SCF (the leftward triangles in Figure 2) is underestimated even at aug-cc-pVQZ; This was a demonstrative example18 which forces the present author to realize that the LP SCF for molecular interaction is a deficient procedure and that the further development of the theory is necessary for the method to be applicable for real chemical problems. The second order correction (the rhombi in Figure 2) converges close to the CP corrected energy at aug-cc-pVQZ, but a slight difference was always found in the previous paper22 as in Figure 2. This difference at aug-cc-pVQZ persists for several clusters examined, and it suggests that the error might be inherent in the second order SPT, and that the higher order perturbation theory might make correction of this slight error. As is seen in Figure 2, this prediction turns out to be correct. The third (downward triangles) E(3) and fourth (upward triangles) E(4) agree with the CP corrected within 0.1 kJ/mol for aug-ccpVTZ and aug-cc-pVQZ. As is mentioned in the analysis of Figure 1a, no difference is found for the filled (use of ALExMO only) and the unfilled (use of all MO) points for these large basis sets. On the other hand, it is notable that both upward and downward unfilled triangles for cc-pVDZ almost coincide with the uncorrected SCF energy; As in Figure 1c, if all of the excited orbitals are used in the perturbation expansion (η ) 0), the BSSE is reintroduced. So the use of all orbitals in the perturbation expansion for the smaller basis sets is not recommended. The restriction to use only the ALEx MOs is too severe for these small basis sets, and E(3)(0.99999) and E(4)(0.99999) tend to underestimate the binding energy. As a compromise, a part of the delocalized excited MOs may be included in the expansion by setting η ) 0.99 or 0.90. Such calculations are shown in Figure 3. Another finding in Figure 2 is the nearly parallel relation of the LP SCF and E(4)(0.99999) plots if the point for cc-pVDZ at the left end is excluded. The difference of two plots ranges from 2.6 to 3.1 kJ/mol for HF dimer and from 3.6 to 4.4 kJ/mol for water dimer. This difference is the stabilization energy gained by mixing the LP SCF configuration and the charge-transfer (CT) excited configurations, and therefore it corresponds to the CT contribution to the binding energy. The energy decomposition of the molecular interaction is still fashionable (see reference,21 and the other references are therein). The present paper is not intending to discuss the energy decomposition, but it can be emphasized that the CT stabilization is caused by the configuration mixing of the charge-transfer from the absolutely local occupied MO to the absolutely local excited MO and that this configuration mixing in the third and fourth order terms cannot be transformed to a form of the orbital mixing. The difference of two plots is slightly basis-set dependent, but not much. The CT contribution to the Hartree-Fock binding energy ranges from 18% to 22% for the HF dimer and from 26% to 30% for the water dimer.

16108 J. Phys. Chem. B, Vol. 112, No. 50, 2008

Iwata setting η ) 0.90 induces the enough CT energy together with a part of BSSE. The characteristics for basis set 6-311++G(3d,2p) in Figure 3b is very similar to those for basis set aug-cc-pVDZ; the number of the diffuse functions is the same in both basis sets though the former has more polarization functions. For the linear isomers of (HF)n the similar η dependence was found, and on the other hand, for the strained cyclic isomers of (HF)n, the slightly different trends are found (those are not shown). More careful analyses are required to have a good balance of the BSSE and the charge-transfer terms. Further Discussion

Figure 3. Binding energy per water for various isomers of water clusters. The basis set used is (a) aug-cc-pVDZ and (b) 6-311++G(3d,2p). Filled circle: uncorrected SCF. Filled square: Counterpoise corrected SCF. Filled downward triangle: E(3)(0.99999). Unfilled rhombus: E(3)(0.99). Filled leftward triangle: E(3)(0.90). Unfilled downward triangle: E(3)(0).

Figure 3 shows the binding energy per water for various water clusters. The geometries of the isomers are determined by B3LYP/6-311+G(d,p) and are taken from the work by Ohno et al.27 The isomers are selected by them to study various types of the hydrogen bonds in water clusters, and they are not necessarily the most stable isomers. For instance, “h2o3” stands for the ring isomer, whereas “h2o3aa9” and “h2o3dd5” are the chain isomers of the middle water being double hydrogen acceptor and double hydrogen donor, respectively. The ordinate is the energy per water, not per hydrogen-bond. The basis set of Figure 3a is aug-cc-pVDZ. As shown in Figure 1b, the third order correction E(3)(0.99999) restricted to ALExMO underestimates the binding energy, whereas E(3)(0) evaluated by using all excited MOs overestimates it. The similar trends are clearly seen in Figure 3a, but the underestimation of E(3)(0.99999) is much more enhanced for the clusters of the stronger and more congested hydrogen bond networks. On the other hand, the overestimation of E(3)(0) is nearly constant over all of the clusters. The unfilled rhombus and filled leftward triangle points are for E(3)(0.99) and E(3)(0.90), respectively. In the figure, the latter points are almost completely overlapped with the points of the counterpoise corrected energy (filled square) and cannot be seen clearly. Because the CT contribution to the binding energy and the BSSE cannot be separated with a finite size of basis set, the partial delocalization of the excited orbitals by

Figures 1 and 2 demonstrate the healthy trends of the single excitation perturbation theory in terms of the locally projected MO (LP SPT). With the enlarged basis sets, the third order corrected binding energy becomes close to the counterpoise corrected binding energy with the cost less than a single supermolecule calculations. The majority of the orbital basis inconsistency (OBI) within the Hartree-Fock level of theory is now corrected by the third order LP SPT E(3). Of course, this is true only for the weakly interacting system, which can include the strong hydrogen bonding system as HF dimer. Now the OBI in the HF level of theory is under control, and the present method must be a good starting point for examining the OBI in the correlated level of theories and the configuration basis inconsistency (CBI). The restrictions imposed on the locality of the excited orbitals and on the types of electron configurations in the perturbation expansion have to be examined both theoretically and numerically. It should also be noted that the use of the extended basis sets does not require a restriction on the excited orbitals; that is, eq 1 does not have to be solved, which might be important in developing the analytical gradient methods for E(2) and E(3). Figure 3 suggests the practical use of the third order LP SPT with a basis set dependent threshold parameter η. This kind of the empirical technique is particularly useful and powerful when it is combined with the use of multilevel basis sets. For instance, in evaluating the total interaction energy, aug-cc-pVQZ is used for the core part of the cluster and all of the excited orbitals are selected in the perturbation expansion, whereas for the outer part, aug-cc-pVDZ is used and only the excited orbitals with η > 0.90 are selected. We can use the technique in the direct Monte Carlo simulation for the intracluster reactions. The application to the density functional theory is straightforward within the one-electron perturbation term V1, as was done by Khaliullin et al.21 But again, care should be taken for the use of the excited orbitals, otherwise the BSSE is reintroduced. Because the third order correction term (7) can be written as occ exct occ exct

E(3) )

∑ ∑ ∑ ∑ ab,t(1)〈(b f t)s|Hˆ - ˆF KS|(a f r)s〉aa,r(1) b

t

a

τ

(12) by replacing the Fock operator Fˆ with the Kohn-Sham operator FˆKS, eq 11 might be applied with the use of Kohn-Sham locally projected virtual orbitals. Acknowledgment. This paper is dedicated to the author’s old mentor and friend, Prof. Karl Freed. Without the collaborative works with Karl in the early 70s, the author could not have switched to become a theoretical chemist. The work is partially supported by the Grants-in-Aid for Science Research (Nos. 17550012 and 20550018) of JSPS. The author is appointed as

Absolutely Local Excited Orbitals a special professor under the Special Coordination Funds for Promoting Science and Technology from MEXT, Japan in 2004-2008. He sincerely thanks Prof. M. Aida of Hiroshima University for giving him an opportunity to return to the research. A part of computations was carried out at the RCCS, Okazaki Research Facilities, National Institutes of Natural Sciences (NINS). References and Notes (1) Castleman, A. W.; Hobza, P. Chem. ReV. 1994, 94, 1721. (2) Brutschy, B.; Hobza, P. Chem. ReV. 2000, 100, 3861. (3) Jeziorski, B.; Szalewicz, K. Handbook of Molecular Physics and Quantum Chemistry; Wilson, S., Ed.; John Wiley & Sons: New York, 2003; Chapter 9, p 233. (4) Kaplan, I. G. Intermolecular Interactions: Physical Picture, Computational Methods, and Model Potentials; John Wiley & Sons: New York, 2006. (5) Liu, B.; McLean, A. D. J. Chem. Phys. 1989, 91, 2348. (6) van Duijneveldt, F. B.; van Duijneveldt-van de Rijdt, J. G. C. M.; van Lenthe, J. H. Chem. ReV. 1994, 94, 1873. (7) Xantheas, S. S.; Burnham, C. J.; Harrison, R. J. J. Chem. Phys. 2002, 116, 1493. (8) Pulay, P. Chem. Phys. Lett. 1983, 100, 151. (9) Mata, R. A.; Werner, H.-J. J. Chem. Phys. 2006, 125, 184110. (10) Hobza, P. Annu. Rep. Prog. Chem. Sec.C 2004, 100, 3. (11) Molpro, version 6, a package of ab initio programs. Werner, H.-J.; Knowles, P. J.; Lindh, R.; Manby, F. R.; Schu¨tz, M.; Celani, P.; Korona, T.; Rauhut, G.; Amos, R. D.; Bernhardsson, A.; Berning, A.; Cooper, D. L.; Deegan, M. J. O.; Dobbyn, A. J.; Eckert, F.; Hampel, C.; Hetzer, G.; Lloyd, A. W.; McNicholas, S. J.; Meyer, W.; Mura, M. E.; Nicklass, A.; Palmieri,

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