4-Currents in relativistic quantum chemistry

INTERNATIONAL JOURNAL OF QUANTUM CHEMISTRY, VOL 45, 573-585 (1993)

4-Currents in Relativistic Quantum Chemistry

JOHN AVERY H. C . Orsted Institute, University of Copenhagen, Copenhagen, Denmark

FRANK ANTONSEN Niels Bohr Institute, Universiv of Copenhagen, Copenhagen, Denmark

IRENE SHIM Danish Engineering Academy, DK-2800 Lyngby, Denmark

Dedication

It is a pleasure to dedicate this paper to Professor Enrico Clementi, whose energy, initiative, and technical brilliance have contributed so much to the development of quantum science. Among the astonishing range of fields to which Professor Clementi has made important contributions is relativistic quantum chemistry, and we are happy to be able to discuss one aspect of this field in the present paper.

Abstract

Some topics in relativistic quantum chemistry are reviewed with special emphasis on 4-currents and 4-potentials. It is shown that, both in molecular quantum theory and in solid-state physics, calculations can include relativistic and magnetic effects by means of 4-currents without an excessive increase in complication, provided that 4-component Dirac spinors are used rather than the Pauli approximation. 0 1993 John Wiley & Sons, Inc.

Introduction

Dirac’s relativistic theory [l-111 provides a very natural and convenient way of treating magnetic effects in quantum chemistry. If

(AeXf, iq5exr) (1) Aexf = P

is a four-vector representing the externally applied electromagnetic potential, then the one-electron Dirac equation can be written in the form

where ko = mc/fi and where the yP’s are 4 X 4 matrices given by

0 1993 John Wiley & Sons, Inc. CCC 0020-7608/93/060573-13

574 AWRY, ANTONSEN, AND SHIM

I = ( ; ;), while the u,'s are the Pauli matrices

(4)

In Dirac's theory, the energy of interaction between the four-component spin-orbital I,$a and an externally applied vector-potential is given by

Here,

is the 4-current due to the electron. The term p a a ~ e x t represents the interaction of the charge density with the external Coulomb potential, while the term -(l/c)jaa * A'"' represents the magnetic part of the interaction.

One can show [1,4] that the second quantized operator corresponding to the electromagnetic interaction can be written in the form

where 6: and b, are electron creation and annihilation operators, respectively, obeying the anticommutation rules [3]:

bfb, + btbf = S,, bJbf + bfbf = 0

bsb, + btb, = 0 . (9) If retardation is neglected (and this is a good approximation for atoms and small molecules), the matrix element urSltu is given approximately by

j ; = e4$sypI ,$t ) . (10) Thus, the second quantized electromagnetic interaction operator has almost the same form as that of the familiar second quantized operator representing the Coulomb interaction between electrons [3]. The only difference is that the magnetic effects have been included by replacing a product of transition densities by a sum of products of transition-current-densities:

(11) l 4

p"(x)p"'(x') - -- jE"(X)jZ(X'). c2 p=l

4-CURRENTS IN RELATIVISTIC QUANTUM CHEMISTRY 575

It follows that in the approximation where retardation is neglected the standard quantum chemistry programs can be modified to include relativistic and magnetic effects by modifying the integral package.

In the second quantized formalism, singly and doubly excited states can be represented, respectively, by

and

where

IAo) = b [ b l . . . bi10) (14) represents the N-electron Hartree-Fock ground state. If we expand t,bs in terms of a set of M basis functions:

a=l

then it follows from (8)-(14) and from the anticommuation relations that Roothaan’s equations [5], generalized to include magnetic effects, can be written in the form

M

1 (Fob - SabEi)Cbi = 0, (16) b=l

where Sab is the overlap matrix

s a b = J d 3 X X J X b (17)

and where the Fock-matrix Fab can be written in the form

p,q=1 s=l

j z b ec(X’a’YpXb)

H& 1 d 3 ~ X J H C X b . (18)

The core Hamiltonian operator H C is the Dirac Hamiltonian:

where 0 uj a, = [ u j ) = i y 4 y j j = 1,2,3.

576 AWRY, ANTONSEN, AND SHIM

Free-particle Solutions

The free-particle Dirac equation can be written in the form

[-iAocu * d + 7'4 - E]Xk = 0, where E is the relativistic energy in units of me2 [l], while

n mc

is the Compton wave length and

A0 E - = 3.861205 X lO-"cm

(23) a

- ax, . a . = -

Equation (22) has two linearly independent positive energy solutions for each value of the wave number k

and two linearly independent negative energy solutions:

where

and

(27) 1 + z A i k 2 1 + ...). The 4-componant spinors multiplying the plane waves in the free-particle solutions are just the columns in the matrix:

1 0 K3 K - 0 1 K+ -K3 K3 K - -1 0 K + -K3 0 -1

4-CURRENTS IN RELATIVISTIC QUANTUM CHEMISTRY 5 77

It is easy to verify that the solutions to the free-particle Dirac equation obey the orthonormality relation:

I d3XXisXk‘st = (2~)~6,! ,6(k/ - k)(l + K 2 ) . (29)

lkansition Current Densities

The transition current densities needed for evaluation of the matrix elements u,,llU can be written in the form

J;b = ec(XTapXb) = 1,2,3

j:b = iec(XjXb) . (30)

If the solutions of the free-particle Dirac equation are used as a basis, we can write the currents in the form

J C L CL ss 9 (31) .ks,k’s‘ = e c ( ~ ) l e i ( k ’ - k ) . ~

where

and where

K: + K - Ki - K3

-Ki + K3 KL + K+

-iK: + iK- iKi - iK3 - i K l + iK+ . I M 2 = ( iK4 - iK3

K i + K3 KL - K- w3=( - K i + K+ Ki + K3

578 AVERY, ANTONSEN, AND SHIM

Fourier Transformation to an Alternative Basis Set

Alternative basis sets can be built up by Fourier synthesis from the free-particle solutions: Suppose that the set of functions f a ( x ) and their Fourier transforms a ( k ) are related by

f a ( X ) = / d3ka(k)eik“

Then, we can define a set of basis functions of the form

X a s ( X ) = / d3ka(k)Xk,(X) 9 (36)

where the functions x t , ( x ) are the free-particle wave functions of equations (24) and (25). This type of basis function can also be written as

where

In terms of the alternative basis, the transition current densities become

jasPbs’(x) P = / d3ka*(k) / d 3 k ’ b ( k ’ ) j r ’ S ’ , (40)

f a ( x ) = / d3ka(k)eik’”, (41)

where the 4-current jYls’ is defined by equations (31)-(34) and where a ( k ) and b ( k ) are Fourier transforms of two basis functions in the set, as indicated by eq. (35).

In evaluating the integrals in eq. (40), it is helpful to remember that if

then

/ d3k’a(k‘”Leik’’X = - i a p f a ( X ) . (42)

Making use of these relations, we obtain the currents


.a l ,b l 1 54 = i e c [ f ; f b + 7 A i { ( a 3 f z ) ( a 3 f b ) + ( a - f i ) ( a + f b ) } ] (43)

and so on. The currents can also be obtained directly from the functions shown in Eqs. (37) and (38).

In Eq. (39), the factor 1 1 5 4 8 64

1 - - Aik2 + - Aik4 - - Agk6 + . . . (44) 2 -- - 2

is nearly equal to unity for values of k that are small in comparison to 1/Ao. Thus, for basis functions that are diffuse in direct space, and, hence, sharply localized in reciprocal space, fa = f a . On the other hand, basis functions that are sharply localized in direct space will have Fourier transforms that are diffuse in k-space; and for such basis functions, f a will differ slightly from fa.

Periodic Systems

Let us consider a crystal for which the direct lattice vectors of the magnetic unit cell are

I = nla l + 11282 + n3a3

g = 27r(mlbl + m2b2 + m3b3)

n, = 0 , 5 1 , 5 2 , ..., (45)

(46)

while the reciprocal lattice vectors of the magnetic unit cell are

m, = 0, + 1 , 5 2 , .. . (The magnetic unit cell may be larger than the chemical unit cell, as illustrated in Fig. 1.) The basis vectors of the direct and reciprocal lattices are related by

ai . bj = S i j i , j = 1,2 ,3 . (47) Since both the scalar potential 4 and the vector potential A are periodic within the crystal, they can be expressed in terms of the Fourier series:

8

A ( x ) = ~ A g e i g ” . (48) 8

The time-independent Dirac equation for an electron moving in the periodic potential A , = (A,iq5) is given by

We can try to represent solutions to (49) as a linear superposition of Bloch functions of the form

&(x) = x Ylgei(k+g).x, (50) 8


where the Fourier coefficients qg are 4-component spinors and where the wave number k obeys the Born-von Karman boundary conditions:

N = N1N2N3 being the number of magnetic unit cells in the crystal. Substitution of ( S O ) into (49) leads to a set of simultaneous equations for the Fourier coefficients qg:

e (52) mc2 g/

The total 4-current, due to all of the charged particles in the crystal, has the periodicity of the magnetic lattice, and therefore its components can be represented by Fourier series of the form

[AOLY * (k + g) + 7 4 - E]'P~ + - [+g/ - (Y * Ag/]'P(z-g') = 0 .

Since the time-independent current and the potential are related through Poisson's equation,

the the

v 2 + = -47rp, (54)

Fourier coefficients of the components of the 4-potential are related to those of 4-current by

4 T i?

4% = F P g

The Nearly Free Electron Picture

If we let

(55)

then we can use perturbation theory to solve for the crystal orbitals. The positive- energy eigenfunctions of the unperturbed Hamiltonian HO can be written in the form


Figure 1. This figure shows the space components of the current associated with the antiferromagnetic basis function shown in Eq. (85). al and a2 are basis vectors of the magnetic unit cell. As illustrated here, the magnetic unit cell can be larger than the chemical

unit cell.

while the negative-energy eigenfunctions are

where

In Eqs. (57) and (58), the wave number k obeys the Born-von Karman boundary conditions, (5 l), and the 4-component spinors correspond to columns in the matrix (Y - (K + G ) + y4. The functions ?,Ik+g,s obey


Since

where the integral is taken over the volume of the crystal, (denoted by V), and since

1 + IK + GI2 = lef$,r, (63) the eigenfunctions of Ho shown in Eqs. (57) and (58) obey the orthonormality relation

/ d 3 X d + g , s q k + g ’ . s ’ = as,s’ag,g’ (64)

and the matrix elements of H’ based on the eigenfunctions of HO can be written in the form

where

and

e2 e2 mc2 fic

ro = - = -A 0 = 2.817515 X 10-13cm,

while ( j p & is a Fourier coefficient of the total 4-current in the crystal. (Notice that the periodic perturbation H‘ does not mix basis functions corresponding to different values of k, since k is an index labeling the irreducible representations of the magnetic translational symmetry group of the crystal.) With this notation, perturbation theory yields positive energy crystal orbitals of the form

‘bk+g,s(x) = q k + g , l ( x ) c l -k qk+g,l(X)CZ

z;= 1 ( J J s I s (4Jg -g’ - A- E E (0) (0) qk+gr,st(X) . . * (69)

mc2 8’fg S’ Ek+g,s - Ek+g’.s’

The constants c1 and c2 satisfy Ic1I2 + lc2I2 = 1, and they are chosen in such a way as to make 7,Jk+g,lcl + r]k+g,2C2 an eigenfunction of H‘. The corresponding energies are given by


2

(0) (0) + ... (70) E k + g . s - € k + g ' , s '

+Ex g ' # g s'

We can see that the perturbation treatment results in a small admixture of negative- energy basis functions in the positive-energy crystal orbitals; but since the energies E k + g ! , s ! are negative for s' = 3 and s' = 4, the energy denominators for these terms are large.

(0)

Transformation of the Basis Set

The transition 4-current density and the matrix elements of the periodic perturbation H' can also be expressed in terms of an alternative basis set:

x a , s ( x ) = r ) k + g , s ( x ) a g * (71)

(72)

g

In terms of the new basis, the transition current density becomes j a s v b s ' CL

= , o c ~ a s y r x b s = x a * b g g C L l j g s * g ' s ' ' g , g'

where

and where the components of (Jp)s,sI are defined by Eq. (67). The matrix elements of H0 and of the periodic perturbation H', expressed in terms of the new basis, become, respectively,

and

while the overlap matrix is

J g

As an example of the type of transformation just discussed, we can consider the periodic function

f a ( x ) = x ~ ~ ( x - 1) = x a g e i g . x , (77) I g

where Qa is an atomic orbital and I is a direct lattice vector [Eq. (45)]. Multiplying this function on the left by e ' g ' X / V and integrating over the volume of the crystal, we obtain


Since the integral on the left-hand of Eq. (78) is independent of I and since

where vc = al * (a2 X 8 3 ) is the volume of a unit cell in the magnetic direct lattice, we have

V C ' I * a ( x ) * (80) a, = - d3xe-iB.X

Thus, for example, if

where a0 is the Bohr radius, then

If the atom on which the basis function is localized is shifted by a distance S relative to the comer of the magnetic unit cell, then fa becomes

fb(x) = x * a ( x - 1- 6) (83) 1

and the corresponding Fourier coefficient is changed by a phase factor: v ~ ~ ( x - 6) = age-ig.b. (84) b, = - d3~e- ig .x V C

Figure 1 shows the space parts of the current due to the antiferromagnetic basis function

xai + Xb2 = Iag(r)k+, ,I + e - i " g r ) k + g , 2 ) , (85) g

where k = 0 and ug is given by Eq. (82). This figure illustrates the fact that the magnetic unit cell of a crystal may be larger than the chemical unit cell.

In the present discussion of periodic systems we have not mentioned the effects of exchange. In solid-state theory, these effects are most often treated by means of the Slater potential, and methods for constructing the periodic Slater potential of a crystal have been discussed in two of our previous papers [12,13].

Discussion

Nonrelativistic quantum theory makes use of the scalar potential 4 and the charge density p , whereas magnetic effects are often added in the Pauli approximation. However, special relativity reminds us that neither the scalar potential nor the charge density is a Lorentz-invariant physical quantity: Both are the time-components of 4-vectors.

In this paper we have reviewed some aspects of relativistic quantum theory, placing emphasis on the 4-current as a physical quantity. We hope th2t the discussion given


here indicates that the 4-current and 4-potential can be included in calculations without a great increase in complexity if 4-component Dirac spinors are used in preference to the Pauli approximation. In ab initio molecular calculations, the main change needed is a modification of the integral package, and programs of this type making use of kinetically balanced Cartesian Gaussian basis functions have recently become available [lo, 111. In solid-state theory, the usual methods of calculation (including both the nearly free electron approximation and the tight-binding approximation) can be modified to include 4-currents and 4-potentials without a prohibitive increase in complexity. Calculations of this type in solid-state theory have become especially interesting because of the recent use of polarized neutron diffraction to measure the magnetic structure of crystals.

Bibliography

[ l ] A. 1. Akhiezer and V. B. Berestetski, Quantum Electrodynamics (Interscience, New York, 1965), Chap.

[2] H. A. Bethe and E. E. Salpeter, Quantum Mechanics of One- and Two-Electron Atoms (Plenum, New

[3] J. Avery, Creation and Annihilation Operators (McGraw-Hill, New York, 1976). [4] J. Avery and F. Antonsen, J. Mol. Struct. 26, 69 (1992). [5] C.C. J. Roothaan, Rev. Mod. Phys. 23, 69 (1951). [6] P. Pyykko, Ed., Int. J. Quantum Chem. 25 (1984). [7] 1. P. Grant, B. Gyorffy, and S. Wilson, Eds., The Effects of Relativity in Atoms, Molecules and the

[8] A. K. Mohantry and E. Clementi, Int. J . Quantum Chem. 39, 485 (1990). [9] A. K. Mohanti, F.A. Parpia, and E. Clementi, Chap. 4; F.A. Parpia and A. K. Mohanti, Chap. 5; A.

Mohanti, S. Panigrahi, and E. Clementi, Chap. 15, in Modern Techniques in Computational Chemistry, E. Clementi, Ed. (ESCOM, Leiden, 1991).

[lo] 0. Visser, L. Visscher, P. J. C. Aerts, and W. C. Nieuwpoort, Theor. Chim. Acta 81,405, (1992); [bid., J. Chem. Phys. 96, 2910 (1992).

[ l l ] K. G. Dyall, P. R. Taylor, K. Faegri, Jr., and H. Partridge, J. Chem. Phys. 95, 2583 (1991). [12] J . Avery, P. Sommer-Larsen, and M. Grodzicki, in Local Density Approximations in Quantum

[13] J . Awry and P. Sommer-Larsen, in Density Matrices and Density Functionals, R. Erdahl and V. H.

2.

York 1977).

Solid Stare (Plenum, New York, 1990).

Chemistry and Solid State Physics, J. P. Dahl and J. Avery, Eds. (Plenum, New York, 1984).

Smith, Eds. (Reidel, Dordrecht, 1987).

Received June 1, 1992 Revised manuscript received September 23, 1992 Accepted for publication October 12, 1992

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4-Currents in relativistic quantum chemistry