A mean-field spin-orbit method applicable to correlated wavefunctions

29 March 1996

CHEMICAL PHYSICS LETTERS ELSEVIER

Chemical Physics Letters 251 (1996) 365-371

A mean-field spin-orbit method applicable to correlated wavefunctions Bemd

A . H e g a, C h r i s t e l M . M a r i a n a, U l f W a h l g r e n b, O d d G r o p e n c

" Institute of Physical and Theoretical Chemistry, University of Bonn, Wegelerstrafle 12, D-53115 Bonn, Germany b Institute of Physics, University of Stockholm, Box 6730, S-11385 Stockholm, Sweden c Institute of Mathematical and Physical Science, University of Troms¢, N-9037 Troms¢, Norway

Received 29 November 1995; in final form 1 February 1996

Abstract

Starling from the full microscopic Breit-Pauli or no-pair spin-orbit Hamiltonians, we have devised an effective oneelectron spin-orbit Hamiltonian in a well defined series of approximations by averaging the two-electron contributions to the spin-orbit matrix element over the valence shell. In addition the two-electron integrals were restricted to comprise only one-centre terms. The validity of these approximations has been tested on several palladium containing compounds. Excellent agreement of the matrix elements of the mean-field operator with corresponding full results is observed; deviations amount to a few cm -1 in absolute value or at most 0.2% on a relative scale. The newly defined mean-field operator can thus safely be employed to evaluate spin-orbit effects in transition metal containing compounds.

1. Introduction

Relativistic two-component operators can be obtained from the four-component Dirac-Coulomb or Dirac-Coulomb-Breit equation by means of the Foldy-Wouthuysen (FW) [ 1 ] or the Douglas-Kroll (DK) [2] transformation. In general, the operators of the FW transformation are too singular to use in a variational calculation [3], so that their use is restricted to low-order perturbation theory. By contrast, the operators obtained from the DK transformation do not suffer from singularities and are amenable to a variational calculation. It is practical to separate operators that do not explicitly depend on the spin and take care of kinematical relativistic effects. Moreover, we obtain an an additive spin-dependent operator comprising spin-orbit and spin-spin interactions. It should be emphasized that this separation is possible

in a rigorous manner for the Dirac-Coulomb equation as well [4] and is thus not an approximation related to the transformation to two components. In any case, it may be used to define a one-component (scalar) relativistic approximation, resulting in onecomponent equations of the type used in the usual non-relativistic approach. Spin-orbit coupling can either be neglected (which is often a good approximation for singlet ground states well separated from the rest of the spectrum) or be treated by means of perturbation theory [ 5 - 8 ] or spin-orbit CI [ 9 - 1 2 ] . At present, quantum chemical calculations are routinely carried out using spin-free no-pair operators obtained from the Douglas-Kroll transformation. The spin-dependent operators are more difficult to handle, however, than the spin-independent ones. While most of the relativistic effects at the spin-free level are well described using only one-electron terms [13] this is

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B.A. Heft et a l . / C h e m i c a l Physics Letters 251 (1996) 365-371

not true for spin-orbit coupling. Blume and Watson [ 14,15] in detail investigated the relative size of the contributions of the full microscopic spin-orbit Hamiltonian, the bare one-electron contributions and the part of the microscopic spinorbit Hamiltonian that may be represented as an effective one-electron operator including the screening by the two-electron terms, which may be expressed by some effective nuclear charge for the atoms in a molecule: ../eff so =

e2h ~ 2m2c----g

~

ZiTla t .

--

' . si .

(1)

Here, Z eff denotes the effective charge of the ath nucleus, r,,i the distance between the a~th nucleus and the ith electron, lai the electronic angular momentum operator with respect to this nucleus and s i the appropriate spin operator. In fact, for very light elements the two-electron contribution to the spin-orbit splitting is of the same order of magnitude - but of opposite sign - as the one-electron spin-orbit term [ 14,5,6,8]. Its relative importance to valence-shell spin-orbit properties decreases with growing principle quantum number, but even for third-row transition elements the twoelectron term contributes about 20% of the total matrix element and cannot simply be neglected in general. In the papers mentioned above, Blume and Watson employed a one-determinant ansatz and rigorously separated all terms of the microscopic spin-orbit Hamiltonian which may be represented as an effective one-electron operator from the residual two-electron contributions, which in turn differ for the various atomic configurations and are caused by interactions between electrons in the outer valence shells. They evaluated the operators in an approximation based on Russell-Saunders (L-S)-coupled single determinants. This approach yields atomic spin-orbit splittings in good accord with experiment for the 2p and 3d elements; for the 3p and 4p elements the splittings were considerably underestimated with this method [ 15]. Parts of the errors were believed to be caused by the use of a non-relativistic Hamiltonian in the Hartree-Fock calculations. Dolg et al. [ 16] extended the Blume and Watson approach for use in combination with a quasi-relativistic Hatree-Fock scheme and indeed obtained satisfactory results for the 4f ion Yb +. Another source of errors is electron correlation

which cannot be taken into account within the Blume and Watson scheme but is essential for the correct description of reactions in which transition elements are involved. We emphasize that the approximations in Blume and Watson's approach are not in the treatment of the one- and two-electron contributions (which will be the topic of the present paper), but rather, Blume and Watson discuss the importance of the two-electron terms (in particular in the valence shell), which they evaluate without approximation. They introduce, however, approximations by neglecting non-diagonal terms in an L - S coupling scheme (which is exactly equivalent to j j coupling, if non-diagonal matrix elements are not neglected, and thus one of the two alternative ways to approach intermediate coupling) and, moreover, by invoking a single-determinant ansatz. In the present paper we set out to define an approximate spin-orbit operator, which is well controlled and has a clear relationship to the full microscopic spinorbit operator. It thus retains the spirit of an 'ab initio' method in that a series of controlled approximations to the 'full theory' is introduced, the 'coarsest' approximation being designed in a way that reliable results can be obtained with feasible computational effort for a given system. If desired, the next 'finer' approximation can be used to assert the validity of the coarser one. In our case, we design an approximation such that spin-orbit coupling can be introduced with moderate effort in a CI procedure, realizing intermediate coupling wave functions after diagonalization of the Hamiltonian. We define an effective one-electron spin-orbit operator which includes both the direct and the exchange interactions between the outer valence and the core electrons, but handles the screening of the one-electron terms by the two-electron contributions by means of an average over the electrons (more specifically: the spin couplings) which are active in a configuration interaction procedure. In other words, we introduce a mean-field approximation for an effective one-electron spin-orbit Hamiltonian.

2.

Theory

The Breit-Pauli spin-orbit Hamiltonian has the following form:

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B.A. Heft et al./Chemical Physics Letters 251 (1996) 365-371

7-[S0--e2h

2m2c2

[~i ZZa$ifgia •

a

\r3a ×

Pi

Hmean field

)

ij

= (ilT~so(1) [j)

1

+2 Z

nk{(iklT-lS°(l'2)lJk) k

-- z frij x pi)'(siq-2sj)] i*j \ r3ij

fixed {nk }

- (iklT-/s°(1,2)Ikj) ~- ET-(SO(i) + E T - ( S O ( i , j ) , i

(2)

i,j

Z, denoting the bare nuclear charge of the ath nucleus, Irql = }ri - rjl the distance between electrons i and j, and Pi the momentum operator for the ith electron. The only difference between the Breit-Pauli Hamiltonian and the corresponding no-pair Hamiltonian is the presence of kinematic factors bracketing the microscopic one- and two-electron operators regularizing the interactions at small r. The no-pair and the Breit-Pauli operators have the same structure, and the formulas given below apply equally well to both. The matrix element of the spin-orbit operator between a pair of Slater determinants differing by a single valence spin orbital excitation i --~ j is given by nS,.,0 = (ilT_(SO(l)lj) + 2l E

nk { (iklT-/s°(1,2)

[jk)

k

- (iklT-(s°(1,2)

Ikj) -

(kil~s°(1,2)

Ijk) } , (3)

where 7-/s° ( 1) and ~ s o ( 1 , 2 ) are taken in their BreitPauli [17] or no-pair forms [18], nk denoting the occupancy of orbitals common to the determinants on the left- and right-hand side. In the independent-particle model, Eq. (3) describes valence electrons (orbitals i and j giving rise to the corresponding charge distributions) moving in a field generated by the electrons in orbitals k (which of course includes the valence space). In other words, Eq. (3) defines a matrix element of a Fock operator tor a one-determinant approximation with a certain occupancy, yielding upon diagonalization a HartreeFock state in intermediate coupling. Based on this observation, we define an approximate spin-orbit operator by

-

(kilT-/s°(1, 2)

[jk)}

(4)

and fix the occupation numbers by, e.g., averaging over the two-electron contribution of the valence shell in Eq. (3), introducing occupation numbers between 0 and 1 in the mean field, i.e., the two-electron part in Eq. (3). In this procedure we average over the spin components as well, i.e., we work in a spin-restricted representation. Typically, occupation numbers are chosen as p/m, if the valence space comprises p electrons in m orbitals. Similar averaging procedures have been used in ordinary Hartree-Fock theory, for example Slater's hyper-Hartree-Fock equations [ 19], the grand canonical HF theory [20] or Zerner's configuration averaged Hartree-Fock [ 21 ]. In general, a mean-field approximation is defined by any set of occupation numbers {nk} by means of a corresponding Fock operator matrix element, and the dependence of the results on the specific set of occupation numbers remains to be checked in a practical calculation. We anticipate a weak dependence on the valence-shell occupancy, since the dominating contributions to the screening come from interactions with the core orbitals. Although Eq. (4) defines an effective one-electron operator, the number of atomic spin-orbit integrals which must be calculated has not been reduced. However, the short-range property of the spin-orbit twoelectron operator makes it also a reasonable approximation to discard all multi-centre two-electron spinorbit integrals. In this approximation the effective twoelectron part of the spin-orbit operator depends only on one-centre integrals which can be easily evaluated. Walker and Richards [22] have suggested that if two-centre two-electron integrals are discarded, a more balanced description of the spin-orbit effect is obtained if only one-centre contributions to the oneelectron spin-orbit integrals are taken into account. However, it is not clear if the cancellation of twocentre one-electron terms and two-centre two-electron terms observed by Walker and Richards holds in general, and experience with the hyperfine operator [ 23 ] lends a certain support to an approximation keeping

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B.A. Heft et al./Chemical Physics Letters 251 (1996) 365-371

all one-electron integrals. It is this one-electrontwo-centre/two-electron-one-centre mean-field approximation, which we shall use subsequently. The performance of our mean-field spin-orbit operator is tested by means of all-electron calculations including electron correlation effects on several states of the palladium atom and of PdCI and Pd2~.

3. Computational details 3.1. Basis sets The palladium atomic basis set consists of ( 17s 13p9d) primitive Gaussian type orbitals (GTOs) [24] contracted to [12s9p5d] using a segmented contraction scheme. The contraction coefficients were determined in atomic relativistic no-pair selfconsistent field (SCF) calculations on the d9sl,3D state of Pd. On chlorine we used a primitive (1 ls 7p) of Huzinaga [25] where we have changed the smallest p exponent to 0.25 and contracted to [6s 4p] again using a segmented contraction scheme.

3.2. Calculations All calculations - including the atomic ones - were carried out in a Cartesian representation. Orbitals and wavefunctions were determined using an explicit spinfree external field no-pair operator [26,27] in the Hamiltonian. Spin-orbit coupling is taken into account perturbationally in a separate step. The spin-orbit integrals were calculated using both the Breit-Pauli form (Eq. ( 2 ) ) and the Douglas-Kroll form of the spinorbit operators. As test cases for studying the influence of different choices of the valence occupation in the definition of the mean field on spin-orbit matrix elements we used the diagonal matrix element of the d9s I 3D state of the palladium atom and its off-diagonal interaction with the corresponding ID and the dSs2 3F~ diagonal matrix element. All atomic calculations were carried out in D2h point group symmetry. In order to investigate the effect of neglecting the two-centre two-electron spin-orbit integrals we calculated the first-order spinorbit coupling of the 2A state of the PdCI radical at a bond distance of 3.0 ao and the lowest 2Au state of the Pd + molecular ion. In Pd2 + the internuclear separation

was set to 4.7 a0, slightly shorter than the equilibrium bond distance in the 31£u state of Pd2 [28] because a stronger bond is expected for the molecular ion than tbr the neutral diatomic molecule. Since there were indications that the d shell hole in Pd2~ might localize on one of the atoms for the calculations on Pd + both D2h and C2v point group symmetries were used. Single reference configuration interaction (CI) calculations were performed for the (L z) = +2 components of the d9s I (3D), d9s ! (ID), and d8s 2 (3F) atomic states of the palladium atom using the d9s I (3D) SCF orbitals as the one-particle basis; the (Lz) = 4-{1,3} components of dSs 2 3F required 3 reference states in the chosen Cartesian representation. Complete active space SCF (CASSCF) orbitals of the 2~+ state of PdCI with three electrons occupying 4 bonding and anti-bonding o- molecular orbitals (MOs) were chosen as the basis for a multi-reference single and double excitation CI (MRD-CI) treatment [29-31 ] of the 2A state of PdCI. For Pd~-, an SCF wavefunction comprising a hole in the valence O'u orbital was optimized in order to obtain molecular orbitals for the subsequent MRD-CI step. Finally, spin-orbit matrix elements were evaluated employing the corresponding CI vectors. Mean-field orbitals were defined from the d9s I (3D), d9s I ( I D ) , and d8s 2 (3F) atomic states. The occupation numbers for the D states are n4d = 1.8, n4s = t.0, and for the 3F state n4d = 1.6, n4s = 2.0,

The programs STOCKHOLM [32], MRD-CI [33,34], and the Bonn spin-orbit codes [8] were used in the calculations.

4. Results In the present investigation we have calculated two spin-orbit matrix elements for the D states of the palladium atom, (3DI3D) and (3DIID), and three matrix elements in the 3F manifold. Both D-state matrix elements, calculated by the full Breit-Pauli operator, are of the order of - 1 3 5 0 cm - I . The two-electron contribution to the spin-orbit matrix elements is about 500 cm -1, i.e., 37%. The mean-field operators were defined for three atomic configurations: (d8s2), (d9s I ) and (dl°s°). Matrix elements calculated with all mean-field operators are shown in Table 1. Two properties of the meanfield method can be addressed already at the atomic

B.A. Heft et al./Chemical Physics Letters 251 (1996) 365-371

369

Table 1 Influence of the mean-field approximation on the spin-orbit matrix elements of various electronic states of the palladium atom Approximation introduced

spin-orbit matrix elements ( c m 3D(sld9), ID(sld9), 3F(s2d8)

1)

(3DaI3D8)

(3DdIIDs)

(3F~r [3FTr)

(3F613Fs)

(3F& [3F,b)

full one- and two-electron Breit-Pauli Hamiltonian

-1363.46

-1341.45

-719.57

-1435.67

-2147.24

mean field ( s2d 8 ) mean field (sld 9) mean field ( d HJ)

- 1362.73 -1361.78 - 1360.84

- 1340.84 -1339.90 - 1338.96

-717.50 -717.05 -716.58

- 1431.38 -1430.42 - 1429.47

-2140.49 -2139.02 -2137.54

level: the accuracy and the state sensitivity. Concerning the latter property, the mean-field operators are hardly sensitive at all to the choice of the valence state for which the mean field was defined. The matrix elements in the D manifold, calculated with the different mean-field operators, all agree to within 2 cm -1 . The accuracy with respect to the full Breit-Pauli calculation is very high, within 3 cm -l for all matrix elements between D-states. The error is slightly larger for the (dSs2)3F diagonal matrix elements (max 10 cm -t in 2150 cm - l ) , which still can be regarded as very small, in particular on a relative scale. Tables 2 and 3 show results obtained for the PdCI radical and the Pd~- molecular ion, respectively. Only the molecular 2A state with an open d8 orbital was used in the definition of the mean field for the PdC1 calculations. The spin-orbit matrix element obtained for the 2A state of PdCI using all spin-orbit integrals and the full B reit-Pauli operator is - 1374.72 cm-1, very close to the -1363.46 c m - l obtained for the atomic (3DI3D) matrix element. This similarity implies already that the state is characterized by a fairly localized hole in the Pd 4d~ orbital and that the contribution of CI to the spin-orbit coupling is small. Neglecting the twocentre two-electron integrals gives values for the matrix elements of -1376.76 cm -1, only 2 cm - t larger than the full value. The effect of the two-centre twoelectron integrals is thus negligible in this case. Again, the mean field approximation changes the matrix element by about 2 cm -1, yielding a matrix element of -1374.93, very close to the value obtained from the full operator. Thus, the errors introduced by the twoelectron-one-centre approximation cancel those of the

mean-field approximation in this case. The difference between the corresponding numbers obtained with the spin-orbit operator obtained from the Douglas-Kroll transformation is about the same. However, as mentioned above, PdC1 might not be a good test example for the importance of these integrals since the perturbation induced by the chlorine atom on the (3DI3D) matrix element on Pd is small anyway. A more significant test case for checking the validity of the one-centre approximation is a molecule consisting of two second-row transition metal atoms, as e.g. Pd~-. Again, the error introduced by the proposed approximations is below 2 cm -1 . In general, we find that variations in the CI procedure introduce variations in the matrix element of the same order of magnitude, and that a localization of the hole in C2v symmetry has a much larger influence ( ~ 30 c m - l ) on the matrix element. Thus, we find that the errors involving the meanfield approximation and the neglect of two-centre twoelectron integrals are negligible. If the error should become larger for other compounds, it is possible to improve the calculations by using a resolution of the identity for the mean-field operator. This would simply amount to calculating the two-electron atomic spinorbit integrals in a larger basis set and projecting the contribution to the atomic integrals onto the valence basis set.

5. Conclusions The mean-field method gives accurate results, independent of the atomic state, for the spin-orbit matrix elements in the systems investigated. The molec-

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B.A. Heft et al. / Chemical Physics Letters 251 (1996) 365-371

Table 2 Effect of the two-electron one-centre and spin-orbit mean-field approximations on spin-orbit matrix elements of the low-lying 2A state of PdC1 Level of treatment

Spin-orbit matrix element (cm - l )

(2~12a> PdCI, non-relativistic wave function, Breit-Pauli spin-orbit operator SO "J-~BP' I-centre 2-electron integrals mean-field, l-centre 2-electron integrals

-1381.79 -1383.71 -1381.85

PdCI, no-pair wavefunction, Breit-Pauli spin-orbit operator full ~' ' BsPo _/so l-centre 2-electron integrals BP' mean-field, l-centre 2-electron integrals

-1374.72 -1376.76 -1374.93

PdCI, no-pair wavefunction, no-pair spin-orbit operator full "Hsn o°- - p a i•r /-(so . l-centre 2-electron integrals i~o- pair ' mean-field, l-centre 2-electron integrals

-1356.89 -1358.92 -1357.10

full

Table 3 Effect of the two-electron one-centre and spin-orbit mean-field approximations on spin-orbit matrix elements of the 2A(u) state of Pd+ Level of treatment

Pd+, D2h symmetry, no-pair wavefunction, BP spin-orbit operator full 7-(ns°o - - p a i.r mean-field ~ snoo - - p a i r l-centre 2-electron integrals mean-field, l-centre 2-electron integrals Pd+, C2v symmetry, no-pair wavefunction, BP spin-orbit operator full .~4so '~BP mean-field 7-LBs°P ' l-centre 2-electron integrals mean-field, l-centre 2-electron integrals

spin-orbit matrix element ( c m - I ) (2AI2A>

-1386.22 --1386.07 -1387.94 -1388.04

-1356.55 -1354.35 -1358.76 -1356.72

Pd+, C2v symmetry, non-relativistic wavefunction, BP spin-orbit operator mean-field _(so l-centre 2-electron integrals BP' mean-field, l-centre 2-electron integrals

-1366.30 -1364.05 -1368.51 -1366.43

Pd +, C2v symmetry, no-pair wavefunction, no-pair spin-orbit operator SO 7-/n(~_pair mean-field 7-Lsn o°- - p a i r l-centre 2-electron integrals mean-field, l-centre 2-electron integrals

-1339.04 -1336.85 -1341.24 -1339.20

full

full

B.A. Heft et al./Chemical Physics Letters 251 (1996) 365-371

ular results show that neglecting all multi-centre twoelectron s p i n - o r b i t integrals introduces an error of 2 cm - I (less than 0.2% on a relative scale) for the matrix elements o f PdC1 and Pd~-, which is negligible for both systems. It can thus be safely concluded that the mean-field method, i n c l u d i n g the neglect of multicentre two-electron s p i n - o r b i t integrals, provides a highly accurate m e t h o d for calculating s p i n - o r b i t effects in heavy elements. The m e t h o d differs from the B l u m e - W a t s o n m e t h o d in that B l u m e and Watson extract from the s p i n - o r b i t operator the part which may be represented as an effective one-electron operator, and evaluate it ( a n d the residual two-electron c o n t r i b u t i o n ) in form o f a diagonal matrix e l e m e n t between R u s s e l l - S a u n d e r s states using the W i g n e r - E c k a r t theorem (i.e., the Land6 interval r u l e ) . In principle, no approximations (apart from possibly the restriction to diagonal matrix elements o f single determinantal wavefunctions) are made in the B l u m e - W a t s o n approach. By contrast, we define an effective screening o f ( n o n - d i a g o n a l ) Fock matrix elements ( w h i c h m a y be used to calculate elements of a CI matrix between configuration state functions differing by at most a single excitation) by fixing the occupation defining the mean-field part once and for all, using a suitably chosen configuration to define the occupancy or averaging the contributions o f the valence shell by i n t r o d u c i n g fractional occupation numbers.

Acknowledgement Travel grants by the E u r o p e a n Science F o u n d a t i o n within the R E H E p r o g r a m are gratefully acknowledged.

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