Spin‐dependent unitary group approach to the Pauli–Breit Hamiltonian. I. Firstorder energy level shifts due to spin–orbit interactionsM. D. Gould and J. S. Battle Citation: The Journal of Chemical Physics 98, 8843 (1993); doi: 10.1063/1.464494 View online: http://dx.doi.org/10.1063/1.464494 View Table of Contents: http://scitation.aip.org/content/aip/journal/jcp/98/11?ver=pdfcov Published by the AIP Publishing Articles you may be interested in Spin-dependent electron transport in graphene junctions in the presence of Rashba spin-orbit interaction J. Appl. Phys. 112, 104319 (2012); 10.1063/1.4766812 Convergence of Breit–Pauli spin–orbit matrix elements with basis set size and configuration interactionspace: The halogen atoms F, Cl, and Br J. Chem. Phys. 112, 5624 (2000); 10.1063/1.481137 Spin‐dependent unitary group approach to the Pauli–Breit Hamiltonian. II. First order energy level shiftsdue to spin–spin interaction J. Chem. Phys. 99, 5983 (1993); 10.1063/1.465897 Spin‐dependent unitary group approach. II. Derivation of matrix elements for spin‐dependent operators J. Chem. Phys. 99, 5961 (1993); 10.1063/1.465895 Spin‐dependent unitary group approach. I. General formalism J. Chem. Phys. 92, 7394 (1990); 10.1063/1.458225

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Spin-dependent unitary group approach to the Pauli-Breit Hamiltonian. I. First order energy level shifts due to spin-orbit interactions

. M. D. Gould and J. S. Battlea) Department of Mathematics, The University of Queensland, St. Lucia QLD 4072, Australia

(Received 17 November 1992; accepted 15 February 1993)

In this paper, we investigate the application of the spin-dependent unitary group approach to many-electron systems in the Born-Oppenheimer and Pauli-Breit approximation. The unitary group approach form of the Pauli-Breit Hamiltonian is determined explicitly, which should enhance the usefulness of the spin-dependent formalism in applications. It is applied to determine the first-order energy level splitting due to spin-orbit and spin-other orbit interactions entirely in the unitary group framework. In particular, this work indicates the feasibility of a spin-dependent unitary group approach perturbation theory for the efficient calculation of spin-dependent operators. Our results are also discussed briefly in terms of the unitary group density matrix formalism. The energy level shifts due to spin-spin interaction will be investigated within the unitary group approach in the second paper of the series.

I. INTRODUCTION

It has long been realized that the ability to do accurate electronic structure calculations for medium sized molec­ular systems is one of the ultimate goals of quantum chem­istry. Even with the massive computational difficulties faced when solving the electronic Schrodinger equation for small molecules, the advent of highly efficient numerical methods such as the unitary group approach (UGA) has meant that chemists can now do such calculations. Since the introduction of the UGA into quantum chemistry by Paldus and Shavitt following earlier work of Moshinsky, 1-3 many theoretical and computational advances have been made.4-9 Besides the fact that the UGA automatically pro­vides spin-adapted wave functions, i.e., wave functions which are eigenfunctions of both 82 and 8z , the computa­tion of spin-dependent chemical properties with this scheme should give better numerical results owing to the lack of spin contamination which plagues other ap­proaches. The advantages of the UGA in the spin­independent problem are well known and well docu­mented.4-8

The UGA was originally introduced for spin­independent systems, but recent work1

0--12 on the graphical

spin-dependent UGA shows that this formalism is also ca­pable of handling spin-dependent operators involving com­putational steps fundamentally no harder than those in­volved in the spin-independent case. In fact, an explicit formula for the matrix elements (MEs) of the spin­dependent U(2n) generators in the graphical segment level form of Shavite has been derived in Refs. 11 and 12 (see also Kent et al. 13). Hence it is now possible, particularly in view of the ever increasing processing speed of supercom­puters, to perform spin dependent configuration interac­tion (el) or PT calculations on molecules.

To facilitate the application of this spin-dependent UGA formalism,1l,12 in this paper we determine the UGA

a) Current address: Supercomputer Facility, Australian National Univer­sity, Canberra ACT 2601.

form of the Pauli-Breit Hamiltonian 14-16 and investigate the first order energy level shifts due to spin-orbit interac­tions in the UGA framework. In particular, it is shown that these interactions (to first order in perturbation the­ory) lead to the splitting of the spin-independent energy levels into equally spaced components. We shall also dis­cuss briefly our results in terms of the recently introduced UGA density matrix formalism. 9 In a subsequent paper, we shall also investigate the first order energy level split­tings due to spin-spin interaction which displays features qualitatively quite different to those arising from spin-orbit and spin-other-orbit interactions.

The paper is set up as follows: Sec. II is introductory. In Sec. III, we determine the UGA form of the Pauli-Breit Hamiltonian. In Sec. IV, we consider the resolution of the Pauli-Breit Hamiltonian into spin-shift components of im­portance in a perturbation theory analysis of relativistic energy level shifts. A fundamental role is played by the zero shift component which determines the full Hamil­tonian matrix for wave functions with well-defined total spin and is expected to yield the dominant contribution to the Hamiltonian matrix even when explicitly spin­dependent terms are accounted for. These results are ap­plied in Sec. V to determine the first order energy level splittings due to spin-orbit and spin-other-orbit interac­tions. We as well incorporate the energy level shifts arising from the relativistic spin-independent terms (e.g., the one and two electron Darwin terms, orbit-orbit interaction, and the relativistic mass increase). In Sec. VI, we investi­gate our results in the UGA density matrix framework and conclude in Sec. VII with a brief discussion of our main results.

II. PAULI-BREIT HAMILTONIAN

In an attempt to incorporate some dominant relativis­tic effects into the many body molecular Schrodinger equa­tion, Breit17 proposed an approximate many-electron, rel­ativistic 'Hamiltonian- containing extra relativistic correction terms added on to the sum of independent Dirac

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8844 M. D. Gould and J. S. Battle: Spin-dependent unitary group. I

Hamiltonians. The further approximation of the Breit equation is due to Pauli18 and the resulting operator is now known as the Pauli-Breit Hamiltonian. I 4-16 Through the use of quantum electrodynamics, the Pauli-Breit Hamil­tonian can be shown to be a formal expansion in powers of (V/c)2 of a pseudo-many-electron Dirac Hamiltonian. 14

Though not covariant, this Hamiltonian is considered to be adequate for the description of molecules and various atomic and molecular calculations have shown this to be the case. 15

The Pauli-Breit Hamiltonian for an N-electron M-atom molecule may be expressed as a sum of terms

(1)

where

is the usual spin-independent electronic Hamiltonian with

M eZ V(l)= L II -; II

a=1 rI a

the potential provided by the nuclei. Here we have adopted the standard notation,15,16 in particular (correspondence principle) ,

The second term is a sum of two contributions

2fi2 N M N p4

H 1=21Te

22 L L Za8(rI- Ra)- L ~8 I 2 (3) mec /=1 a=1 1=1 meC

arising from the one electron Darwin term and the relativ­istic electronic mass increase, respectively. Clearly HI de­termines a spin-independent one-electron operator, while H2 is the spin-independent, two-electron operator

e2 ~ [(rI-rJ) (r/-rJ) I 1

H2=-2m~ l:J PJ - IIrI-rJII 3 +lIr/-rJII ·PI

N 1Te2fi2

- L ----yz 8(r/-rJ) I<J mec

(4)

due to orbit-orbit interaction and the two-electron Darwin term, respectively_ 15,16

The remaining (spin-dependent) contributions to the Pauli-Breit Hamiltonian are due to spin-orbit interaction

e2fl N M Za H S=-2 22 L L II R 1138/- [LI-(RaXp/)],

meC /=1 a=1 r/- a

the spin-other-orbit interaction

e2fl ~ (8I +8J )

Hso= 2_2 £.J II -'13 - [(r/-rJ) X (PJ-PI)], mec- I<J rI-rJI

and the spin-spin interaction

(5)

(6)

(7)

the last term being the so-called contact term. Here we investigate the form of the above operators in

the framework of the UGA. This approach was intro­duced 1,2,4 primarily for the solution of the spin­independent molecular Schr6dinger equation which is a formidable problem, even for medium sized molecules_ However, though mathematically convenient, this spin­independent approximation is often physically unrealistic. Indeed, the spin-dependent terms (5)-(7) arising in the Pauli-Breit Hamiltonian are indispensible in areas such as nuclear magnetic resonance (NMR) and electron-spin res­onance (ESR) spectroscopy, 19,20 where they have a natural physical significance. As noted in the Introduction, the extension of the UGA to allow for spin-dependent opera­tors has recently been achieved1

0-12 and preliminary calcu-

.~~ lations on molecular spin densities indicate the feasibility of this approach for large scale spin-dependent UGA cal­culations on molecules. Hence we now briefly discuss the spin-dependent UGA and determine the UGA form of the Pauli-Breit Hamiltonian.

III. UGA FORM OF THE PAULI-BREIT HAMilTONIAN

Following current orbital based approaches to the many-electron problem, we start with 2n orthonormal mo­lecular (or atomic) spin orbitals

liJL> = tPifL(X) , I <;i<;n, JL=±~,

where X= (r,S) denotes the combined spin and spatial co­ordinates. We assume, as usual, that the spin orbitals factor into orbital and spin parts

tPifL(X) = (/J;{r)XfL(S),

where ifJ;(r) are the orthonormal molecular orbitals and XfL are the elementary spin column vectors

X1/2=(~)' X-l/2=(~)' so that XfL(S) =8fLS -

In the second quantized formulation, we introduce corresponding fermion creation and annihilation operators alfL and aifL' respectively, satisfying the familiar anticom­mutation relations

{ at ,a jv} =8ij8fLV

with the remaining anticommutators all vanishing. The space of N-electron states .7t!' N, to which the N-electron wave functions belong, is then given by all Nth-order prod­ucts of fermion creation operators at acting on the physi­cal vacuum state. The unitary group arises naturally in this context since the space of states JF N gives rise to an irre­ducible representation (irrep.) of the spin-orbital Lie group U(2n) with infinitesimal generators

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M. D. Gould and J. S. Battle: Spin-dependent unitary group. I 8845

r;oip,-ata L 1v- ip, jv' (8)

In this formulation of the many-electron problem, a general spin-dependent single-electron operator

N

h= 2: h(I) (9) 1=1

is expressible as a linear combination of U(2n) generators (8) according to

n

h= 2: 2: (i,ulhljv)Et, (10) i,j= 1 p"v= ± 112

where the coefficients are given by the one-electron spin­orbital integrals

(i,u I h Ijv) = J l/J~(xl)h( 1)l/Jjv(Xl)dxl'

Similarly, a spin-dependent two-electron operator

N

g= 2: g(I,J) I<J

(11)

is expressible as a linear combination of the two-electron spin-orbital replacement operators9,21

EUI!-)(kr) =fff;L J<'kr _o~orJ<'ip, (JV) (lp) Jv<" Ip J v<" Ip (12)

according t09,21

1 ~ '" (. k I I' I ) EUp,) (kr) g=- £.. £.. z,u, r g lV, P (jv)(Ip) ' 2 i,j,k,l= 1 p"V,p,T= ± 112

(13)

where the coefficients are given by the two-electron spin­orbital integrals

(i,u,krlgljv,lp) = f l/J~(Xl)l/JtT(X2)g(1,2)l/Jjv(XI) Xl/Jlp(X2)dxl dX2'

In the case that the operators (9) and (11) are spin independent, the spin-orbital integrals above reduce, re­spectively, to

(i,u I h Ijv) = (il h Ij)op,v,

(i,u,krlgljv,lp) = (iklgljl)op,vDrp,

where

(14)

determine one- and two-electron orbital integrals, respec­tively. In this case, the operators (10) and (13) are ex­pressible in terms of the orbital U(n) generators

~= Iff!,. (15) p,

according to

n

h= 2: (ilhlj)E~, i,j=1

(16a)

1 n

g=2 .. 2: (iklgljl)E~~, 1,J,k,l= 1

(16b)

where

(17)

are the orbital analogs of the two-electron replacement op­erators (12).

In particular, the spin-independent electronic Hamil­tonian (2) is expressible in terms of the orbital U(n) gen­erators (15) according to

n 1 n

Ho=.~ (ilj)E~+2 .. 2: (ijlkl)E~~, I,j = 1 I,j ,k,l= 1

(18)

where the coefficients are given by the one- and two­electron integrals (cf. Ref. 1)

(ilj) = fR3 <fof(r) (:~e -eV ) <fo/r) dr,

(ij I kl) =e2 r <fot(rl)<fot(r2) JR3XR3

1 X II II <fo j(rl )<fo/(r2)drl dr2' rl-r2

where, as before,

M Zae V= 2:

a=1 IIr-Rall is the potential provided by the nuc~ei.

Clearly the orbital U (n) generators (15), and thus the spin-independent Hamiltonian (18), commute with the generators

n

E\,= 2: Et" (19) i=1

of the (spin) group U(2). These operators determine the total molecular spin vector in accordance withl,1O

SA EII2 SA E-1I2 SA 1 (E1I2 E-1I2) += -112' -= 112' z=2 112- -112' (20)

where, as usual, S ± =Sx±iSy [note that the operators (20) constitute the generators of the subgroup SU(2) of U(2)]. For such spin-independent problems, it thus suffices to work in a given irreducible representation of the orbital unitary group U(n).

The operators (15) and (19) form the generators of the spin-orbit (SO) subgroup U(n) X U(2) (outer direct product) of U(2n). It is thus appropriate to consider a basis for the N-electron states symmetry adapted to the SO subgroup U(n) X U(2) of U(2n). The resulting spin-orbit basis states may be written

(21)

where, as usual,I,4 (P) designates an electronic Gel'fand­Tsetlin Of Gel'fand-Paldus (GP) tableau labeling the Of-

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8846 M. D. Gould and J. S. Battle: Spin-dependent unitary group. I

bital part of the state and Ms is the azimuthal spin quan­tum number. The states ISMs) constitute the usual basis for the irreducible representations of SU(2) and are eigen­states of the electron number operator

n

N= L ~=L~ JL= ± 112 i=1

with eigenvalue N [and hence are also U(2) states]. The total spin of the states (21) is given by S=b/2, where

p=(abc), N=2a+b, a+b+c=n

are the Paldus labelsl ,4 of the orbital group U(n) [forming the top row of the pattern (P)] which determine the irre­ducible representation of U(n) concerned.

It is worth noting that the U(2n) generators may be expressed in a natural SU(2) adapted form by introducing the operators

S(i,j) =~oPfi'fv, (22)

where 0" denotes the usual Pauli 0" matrices

-i) (1 o ' O"z= 0 ~1) -(23)

The operators (22) are equivalent to those appearing in the work of Kent et alY We note that in Eq. (22) we have adopted the convention, maintained throughout the paper, of summing over repeated spin indices fL and v. In terms of the above operators, the total molecular spin vector is eas­ily seen to be given by (cf. Ref. 9)

n

s= L S(i,i)=~oP~. (24) i=1

The operators (22) satisfy the commutation relations

[Sa(i,j), Sf3(k,I)] = U/2) Eaf3r[SJSr(i,l) +S',sr(k,j)]

1 k i i k +4 Saf3(SjEz-SjEj) ,

from which we obtain, in particular,

[Sa,Sf3(i,j)] =iEaf3,sii,j) ,

where a, (3, y, etc. refer to the Cartesian components of the spin and Eaf3r is the alternating tensor. These commutation relations demonstrate that the operators (22) transform as a rank 1 tensor operator of SU(2) (vector operator) and together with the U(n) generators [SU(2) scalars] form an alternative useful basis for the U(2n) Lie algebra.

To obtain the UGA form of the Pauli-Breit Hamil­tonian, we first note that HI' being a spin-independent one­body operator, is expressible as a linear combination of orbital U(n) generators according to [cf. Eq. (16a)]

n

H I == L (ilhtlj)~, (25) i,j=1

where hi is the one-electron operator [cf. Eq. (3)]

1Te2fi2 M p4 hi =-2 2 2 L Za8(r-Ra) -~8 2' - (26)

mec a=1 mec

while H2 , representing a spin-independent two-body oper­ator, is expressible [cf. Eq. (16b)]

1 n

H 2==- L (iklh2Ijl)E~1, 2 i,j,k,Z=1

(27)

where h2 denotes the two-electron operator [cf. Eq. (4)]

e2 [(r1-r2) (rl-r2) 1]

h2(1,2)=-2meC2P2' II r l-r211 3 +llrl-r2113 'P2

1Te2fi2 -~S(rl-r2)' me~

(28)

The spin-orbit operator Hs being a spin-dependent one-body operator is expressible as a linear combination of U(2n) generators according to [cf. Eq. (10)]

n

Hs== L L (ifL I hsljv)Ef", i,j=1 JL,V=± 112

where hs denotes the operator [cf. Eq. (5)]

~fz M Za hs=~ L Ilr-Rall S· (L-RaXp).

2me~ a=1

We now note that the spin-orbital integral above factorizes into orbital and spin parts since

(ifL IS· (L-RaXP) Ijv) =~~. (il L- (RaXP) Ij),

where, as above, 0" denotes the Pauli spin matrices (23). We thus arrive at

1 n .

HS=-2~' .~ (ilhslj)Efv, /,J=I

(29)

where hs denotes the vector operator

e2fz M Za hs=~ L Ilr-Rall (L-RaXP)·

2me~ a=1 (30)

Alternatively, in terms of the spin-orbital operators (22), we may write

n

Hs= L (ilhslj)' S(i,j). (29') i,j=1

It is worth noting that for many systems, this term ac­counts for the dominant relativistic contribution.

The spin-other orbit operator (6) determines a spin­dependent two-body operator and thus is expressible as a linear combination of the spin-orbital replacement opera­tors (12) as follows [cf. Eq. (13)]:

1 ~ '" (. k Ih I' I )E(iJL) (kT) Hso=-·£.. £.. IfL, T so jV, P (jv)(lp) ' 2 i,j,k,l= 1 JL,V,T,p

where, from Eq. (6),

e2fz (SI +S2) hso= m;c2 I1r l-r211 3 . [(rl-r2) X (P2-PI)]'

In this case, the spin-orbital integral factors into a spin and orbital contribution according to

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M. D. Gould and J. S. Battle: Spin-dependent unitary group. I 8847

= (/L,rl S, +S21 v,p) . (ikl (r,-r2) X (P2-P,) Iii),

where the first matrix element on the right-hand side is expressible using Pauli (J' matrices as

(/L,rl SI +S21 v,p) =~(ats;+O';~).

We thus obtain

where hso now denotes the two-electron vector operator

e2fz 1 hso(l,2) ==u II 113 (rl-r2) X (P2-PI)' (31) mec rl-r2

Using the easily established result

I( .....J.LS11'+ 1'S1J.L)E(iJL)(kT) _1.....J.L(EiJL 1i'k+EiEkJL 2 U'vUp O'p<J'v (jv)(lp)-2U'v jv'""1 j Iv

we thus arrive at

1 n

H so=4 at • .. L (ik lhso ljl)(EYvE7 1,},k,I=1

. kJL k i +EjElv -2oj Ej,,) (32)

which may be expressed in terms of the generators (22) as

1 n

Hso='2 .. L (iklhsoU/)' [S(i,j)E7 1,},k,I=1

. k +EjS(k,/) -2SjS(i,/)]. (32')

The remaining term H ss , corresponding to the spin­spin interaction (7), is more involved although also simple. First, since Hss is a spin-dependent two-body operator, we have the following expansion in terms of the replacement operators (12):

rr 1 ~ ~ (. k Ih I' I )E(i/-L)(k1') .t:lss='2 .. £.J £.J I/L, r ss jV, P (jv)(lp) , 1,},k,l= 1 JL,V,1',p

where hss denotes the two-electron operator

In this case, the relevant spin-orbital integral is a sum of two terms each of which factor into an orbital and spin contribution according to

(i/L,kr I hss Uv,lp) =!at· (ik I hss I if) . 0'; +! at· O';(ikl hcljl),

where hss denotes the dyadic

e2rz2 1 [3(rl-r2) (rl-r2) hss= - m;~ IIrl-r2113 IIrl-r2112

and he is the contact operator

(33)

(34)

The second term above simplifies upon using the easily established relation

at· 0';=2~S~-~S;. (35)

Substituting into the above, we thereby arrive at

1 n rr .....J.L ~ ('kl h I '/) 1'R(iJL) (k1')

.L.l ss ='8U'v· .. £.J I ssJ 'O'p-(jv)(lp) 1,},k,l=1

+ 1 ~ ( 'k I h I 'Z) [2E(iJL) (kT) Eik] '8 .. £.J 1 c J (jT)(lJL)- jl' 1,},k,I=1

Furthermore, from the antisymmetry property of the re­placement operators (12), we have9,21

E (iJL) (k1') _ E(kT) (iJL) _ Eki (j1')(lJL) - - (j1')(lJL) - - jZ

from which we finally obtain

1 n rr ~.....J.L ('k I h I 'Z) 1'R(iJL)(k1') LZss='8 .. £.J U'v' 1 ssJ 'O'P-(jv)(lp)

l,},k,l= 1

(36')

Alternatively, in terms of the SU(2) adapted generators (22), we may write

1 n

Hss='2 .. L SCi,j)' (iklhssljZ) 'S(k,Z) 1,},k,l=1

1 ~ k' 'k '8 .. £.J (ikl hcUZ) (2EA+Ejz) (36') 1,},k,l=1

as shall be demonstrated in the second paper of this series. In particular, we note that the second contribution above, corresponding to the contact term, determines a spin­independent two-body operator.

Equations (25), (27), (29), (32), and (36) thus com­pletely determine the full Pauli-Breit Hamiltonian in terms of the orbital U(n) and spin-orbital U(2n) generators. Further, as written, the above terms are all in a natural SU(2) tensor form, which is advantageous numerically. It should be emphasized that these spin-dependent effects are expected to be small for many systems, owing to their rel­ativistic nature, so that it is still appropriate to adopt the SO basis (21) even when taking into account explicitly spin-dependent operators. Furthermore, the basis (21) leads to a Hermitian block structure for the matrix of the Pauli-Breit Hamiltonian in which we may consider the blocks outside the main diagonal to be a perturbation of a spin-independent problem and thus develop a spin­dependent UGA perturbation theory.

As a start in this direction, we investigate in Sec. V the first order relativistic corrections to the energy arising from spin-orbit and spin-other-orbit interactions. The spin-spin interaction requires an independent treatment since it leads to energy level splittings which are qualitatively quite dif-

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8848 M. D. Gould and J. S. Battle: Spin-dependent unitary group. I

ferent from those arising from the spin-orbit terms inves­tigated below, as will be seen in the second paper of this series.

We first need a resolution of the U(2n) generators into spin-shift components which effect well defined shifts on the total spin 8. This is particularly the case since it is the zero-shift components which determine the (first order) energy level splittings.

IV. SPIN-SHIFT COMPONENTS

Here we consider the resolution of the spin-orbit terms into well defined spin-shift components. The spin-spin in­teraction will be investigated separately elsewhere. The re­maining terms in the Pauli-Breit Hamiltonian are spin in­dependent and are thus automatically diagonal in the total spin 8 (and azimuthal spin Ms).

Following the spin-shift formalism,1O,22 the U(2n) gen­erators are decomposable into three components

(37)

The operators E( -)f., and E( +)f., lower (and raise, re­spectively) the total spin 8 of the state by one unit, and the operators E(O)f., leave the total spin of the state un­changed. This algebraic decomposition leads, in the SO basis (21), to block upper (and lower, respectively) diag­onal matrices of the spin-shift operators and a block diag­onal structure for the zero shift component.

The decomposition (37) is important both theoreti­cally and numerically since it induces a corresponding par­titioning of the spin-orbit operators into shift components

Hs=Hs( -) +Hs(O) +Hs( +),

Hso=Hso( -) +Hso(O) +Hso( +)

which corresponds to the decomposition of their matrices into block lower diagonal, block diagonal, and block upper diagonal matrices, respectively. This is important in a per­turbation theory analysis of the energy level shifts due to spin-orbit interactions. These shift components are given by Eqs. (29) and (32), respectively, with the U(2n) gen­erators replaced by their corresponding shift component. Explicitly, we have

1 n .

Hs(€) ="2 ~ .. ~ <ilhslj)E(€)fv, /,}=1

(38)

(E=O, ±). For the operators H o, HI' and H2 of Eqs. (18), (25), and (27), only the zero-shift components ap­pear due to their spin-independent nature, as noted above. Their matrices are thus block diagonal in the SO basis (21).

The MEs of the spin-shift operators (37) have recently been determined in the framework of the spin-dependent graphical UGA,10-12 which in turn, leads to an efficient algorithm for the MEs of the above spin-shift components.

Of particular interest here are the zero shift components which completely determine the first order energy level shifts due to spin-orbit interactions, as noted previously.

Recall from Gould and Paldus lO that the zero spin­shift components of the U(2n) generators can be written in the form

E(O)f.,=~E~~- [28(8+ 1)] -11::.~iif:" (39)

where - 1 A

~=~-2N~

denote the 8U(2) generators and I::. is the U(n) adjoint tensor operator given by

(40)

where E is the nXn matrix of U(n) generators as defined in Gould and Chandler.22 It is important to note9,10 that in Eq. (39), it is understood implicitly that the second term does not contribute when 8=0.

It follows that the zero spin-shift component of the 8U(2)-adapted generators (22) are given by

I::.i. 1 _ So(i,j)=~~E(O)tv= 2S(8~1) 2 aP~.

On the other hand, since the Pauli-spin matrices are trace­less, we have from Eq. (24)

~aP~=~aP~=S

from which we obtain

s . 28(S+ 1) I::.j. (41)

Substituting into Eq. (38) above, we thereby arrive at the following expressions for the zero spin-shift compo­nents of the spin-orbit and spin-other-orbit operators, re­spectively:

n

Hs(0)=-[28(8+1)]-IS' L <ilhslj)I::.~, i,j=l

1 A n Hso(O) = - [28(8+ 1)] -1"2 S' L <iklhsoljl)

i,j,k,/=1

(42)

where again, we emphasize that in the case 8 =0, the above terms are understood to vanish.

Equation (42) clearly demonstrates the transforma­tion properties of the zero-shift components. It implies that the spin-orbit terms lead to first order splitting of the spin­independent energy levels (in general 2S + I-fold degener­ate) into 2S + 1 equally spaced components, as seen below.

V. SPIN-oRBIT ENERGY LEVEL SPLITTING (FIRST ORDER)

To illustrate the utility of the above formalism, here we consider the problem of calculating the eigenvalues of the spin-dependent operator

(43)

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M. D. Gould and J. S. Battle: Spin-dependent unitary group. I 8849

where Ho is the usual spin-independent molecular elec­tronic Hamiltonian (18) and

(44)

is the relativistic correction term [cf. Eqs. (3)-(6)]. Thus Eq. (43) comprises the full Pauli-Breit Hamiltonian minus the spin-spin interaction.

Here we assume that we have solved the spin­independent Schrodinger equation

from a previous spin-independent UGA calculation and wish to calculate the energy level shifts due to the elec­tronic interaction lIs. Following the spin-free formalism, 1/10 is assumed to be a purely orbital wave function belong­ing to an irreducible representation of the orbital unitary group U(n). Experiment indicates that, for many systems, the changes in the energy eigenvalues due to these relativ­istic effects are small in magnitude in comparison to the spin-independent energy eigenvalues. I

4-16 This allows the use of standard Rayleigh-Schrodinger perturbation theory, for degenerate states, for computin¥ the energy level shifts due to the relativistic interaction Hs.

We recall since the Hamiltonian H o is spin indepen­dent that the eigenvalue Eo is (2S + 1)-fold degenerate, where S is the spin of the state 1/10. Here, for simplicity, we ignore extra degeneracies arising from possible (spatial) symmetry constraints. The Rayleigh-Schrodinger theory gives the result that the eigenvalues for the perturbed op­erator (43) corresponding to a q= (2S + 1) degenerate eigenfunction of Ho can be expressed (to first order) as

E=Eo+Es ,

where Es is an eigenvalue of the qXq matrix of lIs with respect to the q=2S+ 1 unperturbed eigenstates, given ex­plicitly in this case by [cf. Eq. (21)]

(45)

Below it is shown that the matrix of lIs, within this (2S + 1) -dimensional manifold of states, has 2S + 1 distinct eigenvalues all equally spaced.

We note first that the spin-independent contributions HI and H2 to the perturbing operator lIs simply effect a uniform shift of the energy by the amount

and thus has no effect on the first order energy level split­tings. It is thus convenient to write the energy shift as a sum of contributions

where Es is now the eigenvalue of the spin-orbit term

Hs+Hso

on the (2S + 1 ) -dimensional manifold of states (45). Since these states have well defined total spin, the ac­

tion of the spin-orbit operators reduce to their correspond-

ing zero shift components (42). From Eq. (42), the ma­trices of these operators on the manifold of states (45) are thus given by

n

Hs= - [2S(S+ 1)] -IS· 2: (il hslj) (1/10 1 Llijl 1/10)' i,j=1

1 Hso= - [2S(S+ 1)] -1 2 S·

n

2: (iklhsoljl) i,j,k,l= I

x (1/101 (LljE'/+EjLl7- 2BJLl}) 11/10)·

We may therefore write

Hs+Hso=aoS,

where a denotes the vector

with

I n

aso L (iklhsoUI) 4S(S+1) i,j,k,l= I

(46)

(47)

It is easily seen from the Hermiticity condition 10,22

(Llj)t =Ll{, h1=hs , h~o=hso

that the vectors as and aso, and hence a, are all real from which it follows that Eq. (46) determines a self-adjoint operator on the space of states (45). Moreover, the eigen­value spectrum is easily seen to be given by

Es=aMs, -S<,.Ms<"S,

where a = 1 a 1 is the length of the vector a. This shows explicitly that to first order in perturbation

theory, the (2S+ 1)-fold degenerate spin-independent en­ergy level splits into 2S + 1 equally spaced components with energies

E=Eo+EI +E2+aMs , -S<,.Ms<"S.

The spacing between these levels is thus given by

BE=a

which, from Eq. (47), is completely determined by the MEs of the operators Ll~ in the spin-adapted basis (21). We note that to obtain the corresponding first order per­turbed wave function, the MEs of the nonzero shift com­ponents (37) will also be required.

It is suggestive that we rewrite Eq. (46) as

Hs+Hso=f3b 0 S, where b= (1//3) a with /3= (efz)/(2meC) the Bohr magne­ton. We may then regard the vector b as the internal mag­netic field induced by the spin-orbit interactions. Rewrit­ing Eq. (47) in operator form, it follows that the vector operators

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8850 M. D. Gould and J. S. Battle: Spin-dependent unitary group. I

A I bS =13 as=

A I bSO =13 aso

I

2{3S(S+1)

1 n

n

L (iJhsJj)A~, i,j=1

4{3S(S+1) L (ikJhsoJkl)

i,j=1

. k . k k' X (AjE[ +EjA[-2Sj Ai)

determine the internal magnetic fields induced by spin­orbit and spin-other-orbit interactions, respectively. This illustrates the interesting role played by the operators A~ for the determination of internal magnetic properties of molecules.

VI. DENSITY MATRICES

In this section, for completeness, we express our results in terms of the UGA density matrix formalism developed in Refs. 9 and 23. This serves the dual role of highlighting the physical significance of the density operators investi­gated in Ref. 9.

With the notation of Sec. III, we begin by introducing the one-electron density matrix9

n

PI(r;r')~= L cf>r(r)cf>/r')EYv, i,j=1

which is uniquely determined by the charge density oper­ator

n

pHr;r')= L cf>r(r)cf>/r')E~, (48) i,j=1

which is an SU(2) scalar together with the rank 1 SU(2) tensor density

s( . ') _1-.tL (. ')1L PI r,r -2u vPI r,r v (49)

herein referred to as the one-electron spin density. The z component of this vector operator determines the usual molecular spin density.9,19,24 In terms of the spin-adapted generators (22), we thus have

n

pf(r;r') = L cf>r(r)cf>/r')SU,j). (49') i=1

We may similarly introduce the two-electron density matrix9,24

n

P2(rlh;ri ,rD~;= .. L cf>r(r,)cf>!(r2) l,j,k,l= 1

X cf> /ri )(Mr~)E~rJ)'c1~~ and the corresponding two-electron spin density9,24

pf(r,h;ri ,rD =!~[p2(rl,r2;ri ,r~)~;+ P2(rl,r2;ri ,r~)~]. (50)

The two-electron density matrix is determined by the above spin density together with five other tensor densities which are listed in Ref. 9. This includes the two-electron charge density9 [a SU(2) scalar]

(51)

where the summation over repeated spin indices is again assumed above and below. We emphasize that the above densiges are here regarded as operators. The density ma­trices arising from a given molecular wave function are given by the corresponding expectation values of the above operators (cf. Ref. 9).

Following the traditional procedure,16,19 if

N

h= L h(I) [=1

is a spin-independent one-electron operator, then h may be expressed as an integral [cf. Eq. (16a)]

h= f., h(1)p~(ri;rl)drl' r1=rl

Here we adopt the usual convention of operating with h ( 1 ) before setting ri = rl and then integrating after setting ri = rl' Similarly if

N

g= L g(I,J) [<J

is a spin-independent two-body operator, then from Eq. ( 16b) g may be expressed as a two-electron integral

g=~ f f g(1,2)p~(ri,r~;rlh)drl dr2

with the usual interpretation of the integral. l6,19 In this way, the spin-independent expressions of Eq. (16) may be expressed soley in terms of the density matrix formal­ism.9,23

In particular, the spin-independent terms Ho , HI , and H2 arising in the Pauli-Breit Hamiltonian (1 )-( 4) may be expressed in the above density matrix formalism. We now demonstrate that the one- and two-electron spin-density operators (49) and (50) playa similar role in determining the spin-orbit operators.

In fact, following Eq. (29), we see immediately that the spin-orbit operator is expressible as an integral

Hs= f. ' hs (1) ·pf(ri;r,)dr" rl=r1

(52)

where hs is the vector operator of Eq. (30). Similarly, the spin-other-orbit operator is expressible [cf. Eq. (31)]

Hso=~ f f hso(1,2)' pf(ri,r~;r'h)dr, dr2, (53)

where we again invoke the usual interpretation for the above integrals.

In the case where the molecular wave function has well-defined total spin S, the spin density operators (49) and (50) reduce to their corresponding zero-shift compo­nent [obtained by replacing the U(2n) generators with

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M. D. Gvuld and J. S. Battle: Spin-dependent unitary group. I 8851

their zero-shift components (39)]. In such a case, the spin density operators reduce to the simplified form

pf(r;r') =Spf(r;r'), (54)

s( "') SA S( ") Ps rbr2,rl,r2 = P2 rl,r2;rl,r2 ,

where pf and p~ are the normalized one- and two-electron spin density operators defined in Ref. 9; viz.

pf(r;r/ ) =

pf(r;r / ) =

1

2S(S+I)

1

2S(S+I)

n

L ¢r(r)¢/r/)a~, i,j=1

n

L ¢r(rl)¢~(r2) i,j,k,Z=1

x¢/r!>¢Z(r2) (a~E7+~/:i7-2{jJa~). (55)

The integrals (52) and (53) in this case reduce to

which is equivalent to Eq. (42). We see from the above that the one- and two-electron

spin densities are important not only for the determination of unpaired spin distributions in molecules, but also arise naturally in the context of spin-orbit and spin-other-orbit interactions. Other two-electron density operators are dis­cussed in Ref. 9. Their connection with spin-spin interac­tion will be investigated in the second paper of the series.

VII. CONCLUSIONS

We have determined the UGA form of the Pauli-Breit Hamiltonian which should facilitate the application of the graphical spin-dependent UGA formalism 10--12 in molecu­lar CI and PT calculations. Moreover, the explicitly (spin­independent) zero spin-shift component of the Pauli-Breit Hamiltonian was obtained and is expected to yield the dominant contribution to the Hamiltonian matrix. Indeed, to first order in perturbation theory, this component com­pletely determines the energy level shifts due to spin-orbit interactions as seen in Sec. V. The connection of our results with the UGA density matrix formalism9,23 was also dis­cussed briefly.

Our results [cf. Eqs. (46), (47), and (55)] demon­strate the central role played by the U(n) adjoint tensor operator /:i~, introduced in Ref. 10, for the determination of one- and two-electron density matrices and first order en­ergy level splittings. In fact, as shown in Ref. 10, these operators completely determine the MEs of the U(2n)

generators and thus the matrices of general spin-dependent operators. Their MEs have been determined in a useful graphical segment level form in Refs. 11 and 12.

From the point of view of future applications, it is necessary to obtain formulas for the MEs of products of U(n) generators and the operators a~ for the efficient han­dling of two-body spin-dependent operators; viz. spin­other-orbit and spin-spin interactions [cf. Eqs. (32) and (36)]. In Refs. 11 and 12, it was shown that the MEs of the a~ operators are fundamentally no more difficult to handle than the U(n) generators themselves. This indicates the possibility of treating this problem along lines similar to those developed for treating spin-independent two-body operators in the traditional UGA formalism. .

Finally it would be of interest to examine our approach in the framework of the spin U(2) and symmetric SN groups as an alternative to the U(n)-based approach of this paper. For recent work along these lines, see Kent et al. 13

1 J. Paldus, in Theoretical Chemistry: Advances and Perspectives, edited by H. Eyring and D. J. Henderson (Academic, New York, 1976), Vol. 2, p. 131.

21. Shavitt, Int. J. Quantum Chem Symp. 11, 131 (1977); 12, 5 (1978). 3M. Moshinsky, Group Theory and the Many Body Problem (Gordan and Breach, New York, 1968).

4 Lecture Notes in Chemistry, edited by J. Hinze (Springer, Berlin, 1981), Vol. 22.

sB. Roos, Chern. Phys. Lett. 15, 153 (1972); P. E. M. Siegbahn, J. Chern. Phys. 70, 5391 (1979).

6F. A. Matsen and R. Paunz, The Unitary Group in Quantum Chemistry (Elsevier, Amsterdarn, 1986).

7 See, e.g., the papers by J. Paldus and I. Shavitt, in Mathematical Fron­tiers in Computational Chemical Physics, edited by D. G. Truhlar (Springer, Berlin, 1988), Vol. 15.

B J. Paldus, in Methods in Computational Chemistry, edited by S. Wilson and G. H. F. Diercksen (New York, 1992), Vol. 4, p. 57.

9M. D. Gould, J. Paldus, and G. S. Chandler, J. Chern. Phys. 93,4142 (1990).

10M. D. Gould and J. Paldus, J. Chern. Phys. 92, 7394 (1990). 11 J. S. Battle and M. D. Gould, Chern. Phys. Lett. 201, 284 (1993). 12M. D. Gould and J. S. Battle, J. Chern. Phys. (in press). 13R. D. Kent and M. Schlesinger, Phys. Rev. A 42, 1155 (1990); R. D.

Kent, M. Schlesinger, and I. Shavitt, Int. J. Quantum Chern. 41, 89 (1992).

14H. A. Bethe and E. E. Salpeter, Quantum Mechanics of One and Two­Electron Atoms, (Springer, Berlin, 1957).

ISp. Pyykko, Adv. Quantum Chern. 11, 353 (1978). 16R. McWeeny and B. T. Sutcliffe, Methods of Molecular Quantum Me-

chanics (Academic, London, 1969). 17 G. Breit, Phys. Rev. 34, 553 (1929); 35, 383 (1930); 39, 616 (1932). 1BW. Pauli, Z. Phys. 43, 601 (1927). 19R. McWeeny, Spins in Chemistry (Poly tech, Brooklyn, 1970). 20J. D. Mernory, Quantum Theory of Magnetic Resonance Parameters

(McGraw-Hill, New York, 1968). 21 J. Paldus and B. Jeziorski, Theor. Chirn. Acta 73, 81 (1988). 22M. D. Gould and G. S. Chandler, Int. J. Quanturn Chern. 25, 553

(1984); 27, 787(E) (1985). 23 J. Paldus and M. D. Gould, Unitary Group Approach to Reduced Den­

sity Matrices II (to be published). 24R. McWeeny, Proc. R. Soc. London Ser. A 253, 242 (1959); R.

McWeeny and Y. Mizuno, ibid. 259, 554 (1961).

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