The Effective Valence Shell Hamiltonian Calculations of Spin−Orbit

J. Phys. Chem. B 2008, 112, 16135–16139

16135

The Effective Valence Shell Hamiltonian Calculations of Spin-Orbit Splittings in Small Diatomic Hydrides† Ye Won Chang and Hosung Sun* Department of Chemistry, Sungkyunkwan UniVersity, Suwon 440-746, Korea ReceiVed: July 14, 2008; ReVised Manuscript ReceiVed: August 21, 2008

Recently, the size extensive, ab initio effective valence shell Hamiltonian method for spin-orbit coupling has been suggested. In essence, this effective Hamiltonian method is equivalent to the quasidegenerate perturbation theory. But the difference lies in transforming the original Hamiltonian into an effective Hamiltonian acting within a relatively small valence in the effective valence shell Hamiltonian method. One advantage of the method is that the spin-orbit coupling energies of all valence states for both the neutral species and its ions are simultaneously determined with a similar accuracy from a single computation of the effective spin-orbit coupling operator. Thus, fine structure splittings are predicted for a number of states of each system for which neither experiment nor theory is available. To assess the accuracy of the effective Hamiltonian method more extensively, test calculations are performed for the spin-orbit splittings in the valence states of small diatomic hydrides and their ions. The calculated spin-orbit splittings are generally in good agreement with experiments and with other ab initio computations. I. Introduction Although spin-orbit coupling can be calculated directly by using the four-component Dirac theory, it still remains a formidable task to include the effects of electron correlation into relativistic spin-orbit coupling calculations. A widely used nonrelativistic approach appends the approximate Breit-Pauli spin-orbit coupling operator to the nonrelativistic Hamiltonian, and the spin-orbit coupling operator is treated as a perturbation.1-3 The majority of previous calculations of spin-orbit couplings are based on the variational method, i.e., on configuration interaction (CI) calculations. A serious drawback of the CI method arises when computing off-diagonal spin-orbit couplings between different electronic states because the CI wave functions for different states generally involve different sets of molecular orbitals. Furthermore, the CI methods do not include higher order effects of spin-orbit coupling that arise from offdiagonal matrix elements of the spin-orbit operator between different nonrelativistic eigenfunctions. On the other hand, there have been some perturbation approaches for spin-orbit coupling.3,4 For example Fedorov and Finley report the spin-orbit coupling constants for selected atoms and diatoms using their own spin-orbit multireference multistate perturbation theory.5 These variational and perturbative approaches for spin-orbit coupling have already been incorporated into efficient suites of electronic structure programs, such as GAMESS6 and MOLPRO.7 Recently, Sun and Freed suggested another perturbationbased theory for calculating spin-orbit coupling by extending the effective valence shell Hamiltonian (HV) method, which is based on quasidegenerate many-body perturbation theory, to enable computing spin-orbit couplings.8,9 The effective valence shell Hamiltonian is a highly correlated, size extensive, ab initio method for calculating the electronic energies of molecules. The effective Hamiltonian is obtained by projecting the exact Hamiltonian into the valence space, † Part of the “Karl Freed Festschrift”. * To whom correspondence should [email protected].

be

addressed.

E-mail:

consisting of many quasidegenerate valence states. The eigenfunctions of HV, in principle, provide exact energies for all the valence states. The HV perturbation expansion is identical to the usual multireference, multistate perturbation treatment,5,10 but there exists a large difference between the two methods. Unlike the usual multireference perturbation theory, the effective valence shell Hamiltonian is unambiguously defined within the full valence space, and the matrix elements of HV can be consistently used to describe any state within the valence space. The efficiency and accuracy of the HV method have already been proven and demonstrated with numerous examples.11-20 The most common method of calculating spin-orbit couplings is by adding the Breit-Pauli spin-orbit coupling operator to the nonrelativistic Hamiltonian, and the spin-orbit coupling is evaluated as a first-order perturbation. On the other hand, in our effective Hamiltonian approach, the exact Breit-Pauli operator (A) is also projected into the valence space, and, consequently, we compute all diagonal and off-diagonal valence space matrix elements of the effective spin-orbit coupling operator (AV). In this fashion, the influence of electron correlation is incorporated perturbatively into the effective spin-orbit coupling operator AV, and off-diagonal spin-orbit coupling matrix elements automatically emerge along with the diagonal expectation values.8 The use of HV (or AV) has the following advantages: Once the matrix elements of AV are evaluated, all the spin-orbit coupled valence state energies for the low-lying states of interest are generated with balanced accuracies. The AV effective operator is independent of the number of valence electrons, which implies that the states of the neutral species and its ions are simultaneously calculated with one set of AV matrix elements. In Section II, in order to introduce the effective valence shell Hamiltonian for spin-orbit coupling to unfamiliar readers, we briefly summarize the HV (or AV) method. The details can be found in ref 8. The step-by-step procedure of AV calculations on diatomic hydrides is presented in Section III, which naturally exhibits the characteristics and usefulness of the effective

10.1021/jp806224s CCC: $40.75  2008 American Chemical Society Published on Web 09/18/2008

16136 J. Phys. Chem. B, Vol. 112, No. 50, 2008

Chang and Sun

valence shell Hamiltonian method. Conclusions are provided in Section IV.

NV

V

A

) AVc +

∑

AiV +

i

II. Effective Valence Shell Hamiltonian for Spin-Orbit Coupling The effective valence shell Hamiltonian HV is obtained by projecting the full Hamiltonian into a valence space that is spanned by a prechosen set of valence orbitals. Perturbation theory decomposes the molecular electronic Hamiltonian H into a zeroth-order part H0 and a perturbation V,

H ) H0 + V

(1)

Quasidegenerate many-body perturbation theory (QDMBPT) transforms the full Schro¨dinger equation,

HΨi ) EiΨi

(2)

into the P0 space effective valence shell Schro¨dinger equation,

HVΨiV ) EiΨiV

(3)

for the projection ΨVi ) P0Ψi of the exact wave functions onto the valence space, where Ei represents the exact eigenvalues of the full Hamiltonian, i.e.,

Ei ) 〈Ψi|H|Ψi 〉

) 〈ΨiV|HV|ΨiV 〉

(4)

The Hermitian form of HV is

HV ) P0HP0 +

∑

1 {P (Λ)VQ0[EΛ - H0]-1Q0VP0(Λ') + 2 Λ,Λ' 0 hc + · · · (5)

where hc designates the Hermitian conjugate of the preceding term, and P0(Λ) ) ∑Λ |Λ〉〈Λ| designates the projector onto the zeroth valence space function |Λ〉, and P0 + Q0 ) 1. The space spanned by P0, the valence space, consists of configurations with all core orbitals doubly occupied, vacant excited orbitals, and all possible occupancies of the valence spin-orbitals. To compute a molecular property that is represented by the Hermitian operator A, the above theory may be applied using the perturbed Hamiltonian Htotal ) H + A, where H is the nonrelativistic Hamiltonian as in the above equations. The matrix elements 〈Ψi|A|Ψj〉 may be transformed using the QDMBPT into the matrix elements of an effective valence shell operator AV between the orthonormal valence space eigenfunctions ΨVi of the HV operator,

〈Ψi|A|Ψj 〉 ) 〈ΨiV|AV|ΨjV 〉

(6)

Again specifying that AV be Hermitian and independent of the state Ψi leads to the lowest nontrivial order perturbative expansion,

AV ) P0AP0 +

∑

1 {P (Λ)VQ0[EΛ - H0]-1Q0AP0(Λ') + 2 Λ,Λ' 0 P0(Λ)AQ0[EΛ - H0]-1Q0VP0(Λ') + hc} (7)

where hc designates the Hermitian conjugate of the preceding two terms. Note that the matrix elements of the leading contribution P0AP0 in eq 7 corresponds to the matrix A within the valence space, while the remainder includes “correlation” corrections involving configurations in the orthogonal Q0 space. The energy-independent form of AV can be decomposed as

1 2

NV

NV

∑∑

AijV +

i j(*i)

1 3!

NV

NV

NV

∑∑ ∑

V Aijk + ···

i j(*i) k(*i*j)

(8) where NV is the number of valence electrons, AVc is the constant contribution from the core, AVi is a one-electron effective operator with matrix elements 〈V|AVi |V’〉 in the valence orbital basis set {V}, and so forth. For the spin-orbit operator, the first-order expansion produces the effective spin-orbit operator AV with up to three-electron effective operators AVijk. The projected HV (or AV) is called the effective valence shell Hamiltonian (or AV operator). The algebraic expressions for effective matrix elements of the electron correlation perturbed operator AV in eq 8 are given in ref 8. For spin-orbit coupling, the perturbation operator A is the Breit-Pauli spin-orbit coupling operator that is appropriate for describing systems with moderate or weak spin-orbit coupling. Then it is substituted as the operator A, i.e.,

A)

R2 2

Z

∑ ∑ r3N [rbIN × bpI] · bsI- R2 ∑ ∑ r13 [rbIJ × bpI] · I

N

IN

2

I J(*I) IJ

bJ] (9) [s bI+2s where the indices I and J designate the electrons, N denotes the nuclei, and R is the fine-structure constant. b sI is the spin operator for electron I, b rIN is the position vector of electron I from the nucleus N, b pI is the momentum vector of electron I, b rIJ is the relative position vector of electron I with respect to electron J, rIN is the distance between electron I and nucleus N, rIJ is the distance between electron I and J, and ZN is the nuclear charge of nucleus N. III. Spin-Obit Couplings in Diatomic Hydrides The evaluation of the spin-orbit coupling energy using the AV formalism proceeds as follows: Step 1. We choose the small diatomic hydrides, XH (X ) i, Be, B, C, N, O, F, Na, Mg, Al, Si, P, S, and Cl). The purpose of this work is to calculate fine structure splittings (or level intervals) that are very small, which provides more severe test on the AV formalism. Step 2. The basis sets used are the correlation consistent augcc-pVTZ for BeH, BH, CH, NH, SiH, PH, SH, and HCl, augcc-pVQZ for LiH, OH, and HF, and cc-pVQZ for NaH, MgH, AlH.21 The HV calculations converge upon increasing the size of basis set. The basis set listed above is the smallest basis set that produces the converged value of fine structure splittings for each hydride. The necessary integrals between basis functions, of course, including the orbital angular momentum (lb) integrals are all calculated using the GAMESS package. Step 3. Self-consistent-field (SCF) calculations are performed for the ground electronic state (1Σ+ for LiH and NaH, 2Σ+ for BeH and MgH, 1Σ+ for BH and AlH, 2Π for CH and SiH, 3Σfor NH and PH, 2Π for OH and SH, 1Σ+ for HF and HCl) to generate a set of molecular orbitals and orbital energies. The SCF molecular orbitals are divided into three groups: the core, valence, and excited orbitals. In general, the choice of valence orbitals is arbitrary, since they are quasidegenerate, to guarantee the convergence of perturbation expansion. For the second row atoms, Li through F, the valence orbitals are the 2s and 2p orbitals, and for the third row atoms, Na through Cl, they are the 3s and 3p orbitals. For H, of course, the 1s is a valence orbital. Therefore for the second-row hydrides, the 1σ is the core orbital. The valence orbitals are 2σ, 3σ, 4σ and 1π.

HV Calculations for Spin-Orbit Coupling The rest of the higher lying orbitals are excited orbitals. For the third-row hydrides, the core orbitals are 1σ, 2σ, 3σ and 1π. The valence orbitals are 4σ, 5σ, 6σ and 2π. Again the rest are the excited orbitals. The collection of valence orbitals forms the valence space. For the hydrides having a small number of valence electrons, e.g. LiH, some of the valence orbitals emerge as the virtual orbitals in SCF. We have not attempted to improve these virtual orbitals. Step 4. The Hamiltonian and spin-orbit matrix elements in the atomic orbital basis are transformed to those over molecular orbitals. The b s spin part of the spin-orbit operator is also incorporated to produce integrals in the spin-orbital molecular basis. The calculations of the matrix elements of AV use the spin-orbital basis. The merit of this procedure is to directly yield the total b L ·b s integrals over spin-orbitals. This procedure may appear to be more time-consuming than the customary process. Nonetheless, it makes one generate the symmetry adopted configuration state functions |Ω〉 easily. The spin perturbed CI matrix with these already symmetrized configuration state functions can also be easily set up (see Step 7). Step 5. The matrix elements of the effective Hamiltonian operator HV and the effective spin-orbit operator AV are evaluated by using the second order approximation of eq 5 for HV and the first-order approximation of eq 7 for AV. The secondorder HV has been found to be good enough for calculating the potential energy curves of diatomic hydrides.22-26 The firstorder AV (evaluating the first and second terms in eq 7) is sufficiently accurate when A is small. The second term in eq 7 represents the dynamical correlation correction. For the states reported in Koseki et al.’s theoretical computations,27,28 the present AV calculations are performed at their internuclear distances for the purpose of comparison. For other states including ionic states, the calculations are performed at an equilibrium internuclear distance of the ground-state of neutral hydride. The averaged value of the valence orbital energies is used to guarantee convergence and to simplify the expressions for the matrix elements of the effective operators. Since the main purpose of the work is to show the validity of the HV(or AV) method, the present calculations only use the simple approximate one-electron spin-orbit coupling operator. The customary modification of introducing an effective nuclear charge is not considered. Step 6. Constructed are the spin-orbit symmetry adopted configuration state functions corresponding to | Ω〉 eigenfunctions for diatomic hydrides within the valence space defined in Step 3. The number of configuration state functions depends on the number of valence electrons and Ω, but, in general, it is very small. Therefore the valence CI matrix is very small to easily identify the symmetry of spin-orbit coupled states. For the current valence space consisting of three σ and one π orbitals, the maximum number of the configuration functions is 17 (when Ω ) 0) for two valence electrons. For three electrons, it is 34 (Ω ) 1/2). It is 60 for four electrons (Ω)0), 65 for five electrons (Ω ) 1/2), 60 for six electrons (Ω ) 0), and 34 for seven electrons (Ω ) 1/2). Step 7. The spin-orbit perturbed valence space CI matrix elements (diagonal and off-diagonal) are the sums of HV and AVmatrix elements between the valence space eigenfunctions of HV to include all spin-orbit interactions among the valence states in our calculations. The valence CI matrix is a real symmetric matrix because the matrix elements of AV are evaluated in a spin-orbital basis and because the full spin-orbit symmetrized configuration state functions are used. Finally, the

J. Phys. Chem. B, Vol. 112, No. 50, 2008 16137 TABLE 1: Fine Structure Level Intervals (cm-1) of the Second-Row Diatomic Hydrides hydride

state

R (au)

ΩLTΩHa

this work

LiH

b3 Π

4.9265

LiHBeH

A2Π A2Π a4Π

3.0425 2.5190 2.5190

BH

B2Π a3Π

2.4740 2.2688

CH+

a3Π

2.1469

0T1 1T2 1/2T3/2 1/2T3/2 1/2T3/2 3/2T5/2 1/2T3/2 0T1 1T2 0T1 1T2 0T1 1T2 1T2 2T3 0T1 1T2 1T1 1T0 0T2 2T3 1/2T3/2 1/2T3/2 3/2T5/2 1/2T1/2 1/2T3/2 3/2T5/2 1/2T3/2 1/2T3/2 1/2T3/2 3/2T5/2 1/2T3/2 3/2T5/2 2T1 1T0 1/2T3/2 0T1 1T2 1/2T3/2 1/2T3/2

0.25 0.25 0.16 2.51 0.95 0.95 0.23 6.36 6.37 21.93 21.93 21.38 21.37 1.36 1.36 24.18 24.18 13.82 13.82 13.82 13.82 31.38 0.03 0.78 13.76 13.85 13.89 30.17 25.68 3.04 1.76 85.75 1.15 35.03 35.25 53.83 92.59 94.49 150.90 339.05

23Π 13∆ 33Π 15Π

CH

X2Π a4ΣA2∆ b4Π

2.1163

NH

E2Π F2Π H2Π J2∆ X2Π B2∆ A3Π

NHOH+

X2Π A3Π

1.9785 2.1456

OH HF+

X2Π X2Π

1.8340 1.8918

NH+

2.0220 1.9596

expt.b

theoryc

2.14

0.20 3.15

5.95

5.84

23

19.95

27.95

31.01

77.8

86.27

34.79

34.54

63 83.83

58.20 82.07

139.7 292.85

144.12 307.13

a

Level interval between ΩL (lower state) and ΩH (higher state). Experimental fine structure splittings (see text) from ref 47. c Theoretical ab initio fine structure splittings (see text) from refs 27 and 28. b

CI matrix is diagonalized to generate the eigenvalues, which are the spin-orbit perturbed energies of the valence states. Step 8. Now we examine how well the effective valence shell Hamiltonian method works. Tables 1 and 2 present the AV calculated fine structure intervals for the valence states of the diatomic hydrides and their ions. States with no fine structure splittings are not listed. The state designation is given using the Russell-Saunders coupling scheme, which may not be appropriate for higher states. The states are assigned by consulting previous computations of potential energy curves and spectroscopy experiments. For the second-row hydrides, see refs 22 and 29-39, and for the third-row hydrides, see refs 23 and 40-46. The entries in the Tables are the energy differences (level intervals) in cm-1 between the two adjacent fine structure levels. For example, the entry [0T1 0.28] on the first line of Table 1 is the energy difference between the 3Π1 and 3Π0 states of the LiH b state, i.e., 0.28 cm-1 ) Energy[3Π(Ω ) 1)] Energy[3Π(Ω ) 0)]. Note that the state to the right of the arrow (T) lies higher than the state on left.

16138 J. Phys. Chem. B, Vol. 112, No. 50, 2008

Chang and Sun

TABLE 2: Fine Structure Level Intervals (cm-1) of the Third-Row Diatomic Hydrides hydride

state

R (au)

ΩLTΩH

this work

NaH

b3 Π

4.1952

MgH

A2Π C2Π 14Π

3.1706

AlH+ AlH

A2Π a3Π

3.0073 3.0402

SiH+

b3Π

2.8726

0T1 1T2 1/2T3/2 3/2T1/2 1/2T3/2 3/2T5/2 1/2T3/2 0T1 1T2 0T1 1T2 0T1 1T2 1T2 2T3 0T1 1T2 1/2T3/2 3/2T5/2 1/2T3/2 1/2T3/2 3/2T5/2 1/2T3/2 1/2T3/2 1/2T3/2 3/2T5/2 1/2T3/2 3/2T5/2 2T1 1T0 3/2T1/2 2T1 1T0 1/2T3/2 1/2T3/2 5/2T3/2 3/2T5/2 3/2T5/2 1/2T3/2

1.82 1.76 24.92 1.64 9.67 9.69 74.43 29.05 29.26 97.96 97.97 101.78 101.81 14.50 14.50 108.68 108.74 144.24 7.87 0.74 60.59 60.59 122.54 10.24 128.51 20.28 294.37 8.05 121.92 116.69 193.25 229.33 214.96 378.55 5.90 2.34 11.37 37.98 636.76

c3Π 13∆ 33Π SiH

X2Π A2∆ a4Σ b4Π

2.8726

PH

F2Π I2Π J2Π H2∆ X2Π A2∆ A3Π

PHSH+

X2Π A3Π

2.6588 2.8724

SH

X2Π a4Σ+ 12∆ 22∆ 32∆ X2Π

2.5339

PH+

HCl+ a

2.7121 2.7728

2.1671

142.83

146.68

295.9

296.39

When the internuclear distance of an ion is the same as that of the neutral molecule, the ion state energies are calculated with the same set of HV and AV matrix elements that are determined for the neutral systems. To demonstrate this feature of HV clearly, SiH+ and SiH level intervals in Table 2 are calculated, on purpose, with the same HV (and AV) matrix elements. For the splitting of SiH X2Π, our AV value is 144.24 cm-1, which is comparable to the experimental value of 142.83 cm-1. A similar accuracy is found for SiH+ b3Π, where our AV value is 97.96 cm-1 and the reliable other theoretical value is 95.95 cm-1. The computations for the ions require the trivial effort of diagonalizing the small valence CI matrix for each ion. This feature represents a strong advantage of the effective valence shell Hamiltonian method: one calculation of HV and AV describes all the valence states with balanced accuracy, regardless of the number of valence electrons. Although scant experimental data are available, our calculated values are in good agreement with experiments and other theoretical values. From the comparison of our theoretical splittings with the experimental data of Herzberg,47 we find that, apart from two exceptions (out of 20 comparisons), our values for all the neutral hydrides and ions agree well with experiments. The exceptions are for the A2Π state of AlH+ and the a3Πstate of AlH, where our calculated fine structure intervals are off by ∼35% from experiment. Tables 1 and 2 also contain computations for a number of states whose fine structure splittings are not available from experiment or from prior calculations.

115.71

118.73

V. Conclusions

212 216.5

192.55 218.65

376.96

381.05

648.13

642.90

expt.a

theoryb

35.3

34.31

108 40.2

107.26 41.34 95.95

Experimental fine structure splittings from ref 47. ab initio fine structure splittings from refs 27 and 28.

b

Theoretical

The traditional definition of the fine structure splitting is the spacing between two adjacent Ω fine structure states. When spin-orbit coupling produces more than two Ω states, there exist more than one fine structure splitting. For example, the fine structure states of 3Π are 3Π0, 3Π1, and 3Π2. They yield two fine structure splittings, i.e., 0T1 and 1T2. These two splittings are found to be very similar but not necessarily the same. For example, as shown in Table 1, for the CH+ a3Π state, the two splittings are the same as 21.93 cm-1 for both 3Π0T3Π1 and 3Π1T3Π2 within the numerical error of computation. But for the CH+ 23Π state, the two splittings are slightly different, i.e., 21.38 cm-1 for 3Π0T3Π1 but 21.37 cm-1 for 3Π1T3Π2. Larger differences can be seen, for example, in the three SiH+ 3Π states in Table 2. The order of fine structure levels is, in fact, important and interesting. It is said that a lower Ω state lies lower in energy (normal ordering) when the valence shell is less than half-filled, but a higher Ω state lies lower in energy (inverted ordering) when the valence shell is more than half-filled. This trend, in general, can be found in Tables 1 and 2. But the simple rule does not apply to all the valence states presented. To our knowledge, the detailed order of fine structure levels of small diatomic hydrides has not been reported.

The effective valence shell Hamiltonian method HV for spin-orbit coupling is applied to calculating the fine-structure level intervals of valence states of the second- and third-row diatomic hydrides and their ions. The fine structure intervals are not found to be constant when many fine structure levels exist. The order of fine structure levels is unambiguously presented for all the states under study. Our calculations are in good agreement with experiments and prior computations. The results are very encouraging, although only the one-electron spin-orbit coupling operator is used. The main reason for this success is that the electron correlation perturbation V and the spin-orbit coupling are treated together on an equal footing in the AV matrix elements. Our calculations also provide predictions of fine structure level intervals for excited and ion states for which there are no prior experimental data or computations. It is again found to be one of many advantages of the effective valence shell Hamiltonian method. Once the matrix elements of the effective spin-orbit operator AV are evaluated, only a small valence space CI calculation is required to determine the spin-orbit couplings for all the valence states. This simultaneous treatment of all valence states saves a lot of computer time. The present calculations demonstrate that the effective valence shell Hamiltonian method performs well and will be useful for studying the spin-orbit coupling of larger molecules. Acknowledgment. The authors thank Prof. K. F. Freed for encouraging them to pursue this work further. References and Notes (1) Hess, B. A.; Marian, C. M.; Peyerimhoff, S. D. In Modern Structure Theory, Part I: AdVanced Series in Physical Chemistry; Ng, C. Y., Yarkony, D. R., Ed.; World Scientific: Singapore, 1995; Vol. 2, pp 152-278, and references therein. (2) Marian, C. M. ReV. Comput. Chem. 2001, 17, 99, and references therein.

HV Calculations for Spin-Orbit Coupling (3) Fedorov, D. G.; Koseki, S.; Schmidt, M. W.; Gordon, M. S Int. ReV. Phys. Chem. 2003, 22, 551, and references therein. (4) Teichteil, C.; Pelissier, M.; Spiegelman, F. Chem. Phys. 1983, 81, 283. (5) Fedorov, D. G.; Finley, J. P. Phys. ReV. A 2001, 64, 042502. (6) Schmidt, M. W.; Baldridge, K. K.; Boatz, J. A.; Elbert, S. T.; Gordon, M. S.; Jensen, J. H.; Koseki, S.; Matsunaga, N.; Nguyen, K. A.; Su, S. J.; Windus, T. L.; Dupuis, M.; Montgomery, J. A. J. Comput. Chem. 1993, 14, 1347. (7) MOLPRO, a package of ab initio programs. Werner, H. -J.; Knowles, P. J.; Almlo¨f, J.; Amos, R. D.; Berning, A.; Cooper, D. L.; Deegan, M. J. O.; Dobbyn, A. J.; Eckert, F.; Elbert, S. T.; Hampel, C.; Lindh, R.; Lloyd, A. W.; Meyer, W.; Mura, M. E.; Nicklass, A.; Peterson, K. A.; Pitzer, R. M.; Pulay, P.; Schu¨tz, M.; Stoll, H.; Stone, A. J.; Taylor, P. R.; Thorsteinsson, T. See http://www.molpro.net. (8) Sun, H.; Freed, K. F. J. Chem. Phys. 2003, 118, 8281. (9) Chang, Y. W.; Sun, H. Bull. Korean Chem. Soc. 2003, 24, 723. (10) Fedorov, D. G.; Nakajima, T.; Hirao, K. J. Chem. Phys. 2003, 118, 4970. (11) Sun, H.; Sheppard, M. G.; Freed, K. F. J. Chem. Phys. 1981, 74, 6842. (12) Freed, K. F. Acc. Chem. Res. 1983, 16, 137. (13) Sun, H.; Freed, K. F. J. Chem. Phys. 1988, 88, 2659. (14) Chaudhuri, R. K.; Freed, K. F. J. Chem. Phys. 1997, 107, 6699. (15) Chaudhuri, R. K.; Stevens, J. E.; Freed, K. F. J. Chem. Phys. 1998, 109, 9685. (16) Chaudhuri, R. K.; Majumder, S.; Freed, K. F. J. Chem. Phys. 2000, 112, 9301. (17) Chaudhuri, R. K.; Freed, K. F. J. Chem. Phys. 2003, 119, 5995. (18) Taylor, C. M.; Chaudhuri, R. K.; Freed, K. F. J. Chem. Phys. 2005, 122, 044317. (19) Chaudhuri, R. K.; Freed, K. F. J. Chem. Phys. 2005, 122, 154310. (20) Chaudhuri, R. K.; Freed, K. F. J. Chem. Phys. 2005, 122, 204111. (21) Basis sets were obtained from the Extensible Computational Chemistry Environment Basis Set Database, Version 6/28/02, as developed and distributed by the Molecular Science Computing Facility, Environmental and Molecular Sciences Laboratory, which is part of the Pacific Northwest Laboratory, P.O. Box 999, Richland, Washington 99352, and funded by the U.S. Department of Energy. The Pacific Northwest Laboratory is a multiprogram laboratory operated by the Battelle Memorial Institute for the U.S. Department of Energy under contract DE-AC06-76RLO 1830. Contact David Feller or Karen Schuchardt for further information.

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