The use of quadratic forms in the calculation of

JOURNAL OF MATHEMATICAL PHYSICS 47, 083505 共2006兲

The use of quadratic forms in the calculation of ground state electronic structures Jaime Kellera兲 Departamento de Física y Química Teórica, Facultad de Química, Universidad Nacional Autónoma de México, AP 70-528, 04510, México D.F., Mexico

Peter Weinbergerb兲 Center for Computational Materials Science, Physics Department, Technical University of Vienna, Gumpendorferstrasse 1A, A-1060, Vienna, Austria 共Received 8 May 2006; accepted 19 June 2006; published online 9 August 2006兲

There are many examples in theoretical physics where a fundamental quantity can be considered a quadratic form ␳ = 兺i␳i = 兩⌿兩2 and the corresponding linear form ⌿ = 兺i␺i is highly relevant for the physical problem under study. This, in particular, is the case of the density and the wave function in quantum mechanics. In the study of N-identical-fermion systems we have the additional feature that ⌿ is a function of the 3N configuration space coordinates and ␳ is defined in three-dimensional real space. For many-electron systems in the ground state the wave function and the Hamiltonian are to be expressed in terms of the configuration space 共CS兲, a replica of real space for each electron. Here we present a geometric formulation of the CS, of the wave function, of the density, and of the Hamiltonian to compute the electronic structure of the system. Then, using the new geometric notation and the indistinguishability and equivalence of the electrons, we obtain an alternative computational method for the ground state of the system. We present the method and discuss its usefulness and relation to other approaches. © 2006 American Institute of Physics. 关DOI: 10.1063/1.2229423兴

I. INTRODUCTION

Since the creation of wave quantum mechanics, in 1926, the calculation of the electronic structure of a many electron system, in its ground state, or near it, is an open subject. This central problem, with no exact analytic solution, has been studied from many points of view. When the study is to be used for the analysis of atoms, molecules, or condensed matter a first useful approximation is to solve the many electron Schrödinger equation for the system. This implies the use of the Coulomb potential of a configuration of nuclei as external potentials and of the electronelectron interaction potentials to a desired degree of accuracy. In this paper, after a brief review of the basic problem for a many electron system in a molecule, we start a systematic analysis of the problem using an accurate formulation, compatible with the initial conditions of the calculation, and through a systematic, geometric algebra based, definition of the configuration space, of the external potentials, of the one electron operators for a many electron system and of the electronelectron interaction terms, we arrive at a formal equation for the total energy. From this, using the fact that we are interested either in the ground state or in stationary states near the ground state, we formulate a variational problem from which a set of tractable equations, which self-consistently define the many electron wave function and density, is obtained. Finally we compare our resulting formalism with the more widely used procedures, showing that those methods are contained as special cases of ours.

a兲

Electronic mail: [email protected] Electronic mail: [email protected]

b兲

0022-2488/2006/47共8兲/083505/12/$23.00

47, 083505-1

© 2006 American Institute of Physics

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J. Math. Phys. 47, 083505 共2006兲

J. Keller and P. Weinberger

The basic approach for the calculation of the ground state electronic structures of atoms, molecules, and solids1–3 considers the steady state Born-Oppenheimer approximation: a fixed given configuration of positively charged point-like nuclei, the so-called “structure” of the molecule, cluster, or crystal. The computational procedure involves the simultaneous calculation of: the N electron ground state electronic wave function ⌿N a function of 3N coordinates, the 共ground state兲 electronic density ␳N共x兲 a function of three coordinates, the total electronic energy EN for the N electrons, a set of M 艌 N auxiliary spin-orbitals ␸ 共called the SO兲, and the SO-energy eigenvalues ␧a. A minimum set of N auxiliary functions is required by Pauli’s exclusion principle. The SO and the ␧a are important as far as they are related to the different type of response functions of the electronic system and as such to the set of its spectroscopic properties. The electronic density, wave function, SO, and ␧a are not observable in themselves but, through several spectroscopies, they can be studied as indirect observables. Point-contact spectroscopy and other recent techniques 共see Ref. 4, and references therein兲 are good examples of such tools. The computational techniques can also be extended to time dependent5,6 and to beyond Born-Oppenheimer7 cases. In these approaches the electron is defined as a physical entity which obeys the Dirac equation for the electron’s experimental mass and charge, the nonrelativistic limit of the Dirac equation is given by the Schrödinger equation if the spin of the electron is considered for magnetic interactions and for statistics. Recently we have shown that there is a new, geometrical, starting point for formulating a mathematical theory for the density and for the related wave function,8–12 we call KKW. There is a resulting methodology developed here. In the calculation of the stationary ground state of the many electron 共many indistinguishable equivalent fermions in general兲 wave function and densities a variational approach optimizes the solutions with respect to desirability criteria, in our case, the lowest ground state energy E. The variational procedure provides a suitable set of constants and a set of auxiliary functions. For ¯H ˆ ⌿dV we should consider the computational method M of lowest ground state energy E = 兰⌿ M variation of a functional

␦

冉冋冕

册 冋冕

¯H ˆ ⌿dV − E − ␮ ⌿ M

¯ ⌿dV − N ⌿

册冊

= 0,

共1兲

where the second square parentheses impose the condition that this is an N electron system. Note ¯H ¯ ⌿ = ␳ 共x兲 imply a projection into real space. ˆ ⌿ = E共x兲 and ⌿ that both ⌿ M N The basic formulation can be summarized as follows: to the N equivalent electron system, in real space x 傺 R3, in volume V, obeying the Pauli’s exclusion principle, corresponds an analytical finite non-negative total density function ␳N共x兲 such that the total electronic energy E = 兰␧␳N共x兲dV is well defined in the system’s volume. To fulfill this condition there should exist a many electron analytical square integrable wave function ⌿N共兵xn其 ; n = 1 , . . . , N兲 with 兵xn其 傺 共R3兲N ¯ ⌿ = 兺N ␳ 共x兲 ⫽兺 M艌N␳ 共x兲. such that ␳N共x兲 ⬟兩⌿N兩2 ⫽⌿ u c=1 c u=1 N ␳c共x兲兲 is Notice that two different expansions of the density are used simultaneously: one 共兺c=1 M艌N the description of the N equivalent electrons in the system, the second 共兺u=1 ␳u共x兲兲 describes the shell structure 共M atomic or molecular orbitals兲. This generates a matrix of descriptions: one description, the shell structure, corresponds to the M 艌 N columns and the other, the per equivalent electron contribution, to the N rows of the matrix. We have also indicated that this Hermitian square should be describable as either a sum of non-negative, finite analytical functions ␳c共x兲 = 兩␺c兩2, one equivalent ␺c for each electron in the system, or 共to be able to agree with the Pauli’s principle兲 as a sum of M 艌 N, weighted by wu = 兩bu兩2, spin-orbital ␾u contributions ␳u共x兲 = 兩bu␾u兩2. This ensures that the physical conditions, which must be obeyed by the wave function itself and by the density, are fulfilled. In the case of the many electron 共fermion兲 system all N electrons 共fermions兲 are equivalent. The “system” to be studied is such that no electron can be distinguished by position. This equivalence requires that the density itself should be describable as a sum of N equal densities ␳one electron共x兲 which should be generated by equivalent contributions. That is ␳one electron共x兲 = 1 / N兺u␳u共x兲. The many-electron

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Quadratic forms and electronic structure

J. Math. Phys. 47, 083505 共2006兲

wave function ⌿N共兵xn其 ; n = 1 , . . . , N兲 should be used to compute the total energy and other properties of this ground state of the system. Historically the Hartree-Fock method 共and its complement configuration interaction兲 fulfilled the conditions above. What is needed is to find the analytical, possible complex, function ⌿共兵xn其兲 of a set 兵xn ; n = 1 , . . . , N其 of N coordinates which allows the factorization of a finite non-negative function

␳共x兲 = ⌿†共兵xn其兲Nˆ共x,兵xn其,兵xn⬘⬘其兲⌿共兵xn⬘⬘其兲 傺 R+ 共the real numbers R = 兵R− , 0 , R+其 are either negative, zero or positive兲. Here Nˆ共x , 兵xn其 , 兵xn⬘其兲 = 兺nn⬘␦xnxn⬘⬘␦xxn is a projector from configuration space X into the real space coordinates x. In the KKW theory 共all relevant points are presented in the following兲 the basic idea is that the density appears as a sum of densities and then the wave function ⌿N should both be the square root of the total density function ␳共x兲 = 兺u␳u共x兲 and also provide the square root ␾u of each one of the contributions ␳u共x兲 to the total density. For this we require the use of geometric 共multivector analysis兲 techniques. In fact the problem is similar to that of finding the linear form 共geometric square root兲 d = ae1 + be2 + ce3 + ¯, which corresponds to the quadratic form d2 = a2 + b2 + c2 + ¯ 傺 R+. Here d is the geometric square root of the scalar quantity d2. For a positive definite function ␳共x兲 = d†共x兲d共x兲 傺 R+ an ordinary 共scalar兲 square root d共x兲⫽ † 冑d 共x兲d共x兲 = 冑a2共x兲 + b2共x兲 + c2共x兲 + . . . is not necessarily analytical, whereas the geometric square root function d共x兲 = a共x兲e1 + b共x兲e2 + c共x兲e3 + ¯, through the use of an analytical set ␾ of auxiliary complex functions ␾ = 兵a共x兲 , b共x兲 , c共x兲 , . . . 其 傺 C, can be demanded to be analytical. The Hermitian squares 兵a2共x兲 , b2共x兲 , c2共x兲 , . . . 其 傺 R+. The analyticity property allows that a set of differential ˆ a共x兲 = ⑀ a共x兲, D ˆ b共x兲 = ⑀ b共x兲 , . . . , which incorporates physical and mathematical equations D a b boundary conditions can be found. The geometric square root. There is a freedom to choose for the description of the shell structure the most convenient anticommuting normalized basis set 兵␣u , 兩␣u兩2 = 1; and ␣u␣v = −␣v␣u, u ⫽ v, u , v = 1 , . . . , M 艌 N其. Otherwise, as the electrons are equivalent, we will also need a symmetrized form of d. For this purpose we will simultaneously introduce a second 共one basis element per electron兲 anticommuting normalized basis set 兵␻Si , 兩␻Si 兩2 = 1; and ␻Si ␻S j = −␻Sj ␻Si , j ⫽ i, i = 1 , . . . , N其 and a normalized vector S = 冑1 / N共␻S1 + ␻S2 + ␻S3 + ¯ + ␻NS兲 where our definitions imply that 共冑1 / N共␻S1 + ␻S2 + ␻S3 + ¯ + ␻NS兲兲2 = S2 = 1. The ␻Si , therefore S, are defined to commute with the ␣u. Consider d = 共b1␣1 + b2␣2 + b3␣3 + ¯ + b M ␣ M 兲S, then d2 = ⌺i 兩 bi兩2. There is a double summation: over the basis set 兵␣u其 and over the per electron in the system set 兵␻Si 其. The summations can be interchanged. This is what is called the geometric procedure as used in the following. The algebra of the 兵␣u其 or of the 兵␻Si 其 is a Grassmann-Clifford algebra 共see Refs. 10–12兲. Although there is no need to define a vector scalar-product we will define in the following an equivalent procedure to simplify the reading of the equations. II. GEOMETRIC FORMULATION OF THE CALCULATION OF THE STATIONARY GROUND STATE OF A MANY-ELECTRON SYSTEM

Here we present the formalism for the calculation of the stationary ground state of the many electron 共many equivalent fermions in general兲 wave function and densities and the corresponding variational approach. A. Configuration space and real space

A basic concept in the study of a many-electron system 共N interacting fermions兲 is, from the above-noted considerations, the simultaneous, repeated, use of real space 共the space of the observer兲 for each one of the fermions of the system: configuration space. Then, if x represents a point in real space, it is customary to represent by X = 兵xa ; a = 1 , . . . , N其其 the set of points in the configuration space X for N fermions.

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The volume integral 兰d⍀ when referred to the coordinates of fermion a is denoted by 兰d⍀a. Also 兰d⍀N indicates the integration over all N space replicas 兰d⍀a. In the formal notation to ˆ to denote that repeated space integration for all follow we introduce an integration operator 兰d⍀ N ˆ N electrons is to be performed. Also 兰d⍀N−i to denote that the repeated space integration for all N electrons except the ith is to be performed. The absolute value of the distance between two fermion points xab = 兩xb − xa兩. Here and in the rest of our presentation we use a geometric notation X = 兺 a ␻axa ;兵a = 1, . . . ,N; ␻a␻b = − ␻b␻a,b ⫽ a其,

共2兲

¯ a such that ␻ ¯ a␻ b introducing the per electron operator ␻a and, also, the projection operators ␻ ¯ aX = ␦ab selecting the part of the configuration space which corresponds to electron a; this is ␻ = x a. This construction allows a clear formal definition of the electrons involved in each part of the calculation. Our geometric procedure introduces one feature of the statistics of the fermion system from the beginning because the interchange ␻a ↔ ␻b in the products ␻a␻b for two electrons, in any given expression, will change the sign of the corresponding terms. When the argument of a function is position, for a = 1 for example, we will also use a nonbold notation or the fermion’s numeral, say ␾共x1兲 ⬟ ␾共x1兲 = ␾共1兲, as equivalent argument.

B. Basic principles

The Principles implied without further discussion in this paper are the same as those of the so-called ab initio approaches:1 • The total energy is a functional of the wave function. • The possibility of using the Schrödinger equation for the N electron system. The kinetic energy is the sum of independent-electron-like kinetic energies for each electron. • The ground state of the many electron system corresponds to the lowest total energy. • The Pauli exclusion principle requires that the description for N electrons, included in the wave function, contains the occupancy of at least M
N pseudo-electron orthonormal spin-orbitals ␾u. • The equivalency and indistinguishability of the electrons require that all electrons are equivalently described. • The operators acting on the ␾u are: multiplicative operators 共constant, variable, and self2 consistent variable functions, differential 共the Laplace operator ⵜ共x , in our case兲 and the u兲 ˆ integral operators 兰d⍀u defined earlier. • A variational approach can be used. The electronic structure calculation of a many electron system in the ground state requires then the simultaneous calculation of: the ground state electronic wave function ⌿N, the 共ground state兲 electronic density ␳N共x兲, the total electronic energy EN for the N electrons, a set of M 艌 N auxiliary spin-orbitals 共SO兲, and the SO-energy eigenvalues ⑀a. A minimum set of N functions is required by Pauli’s principle. ¯ ⌿ both quantities, wave function ⌿共X兲 and density ␳共x兲 could be considered as From ␳共x兲 = ⌿ ¯ ⌿. the fundamental variable, provided that derivatives of ␳共x兲 are considered derivatives of ⌿ ¯ , ⌿其 In our theory the density appears as a sum of densities and then the wave functions 兵⌿ should together: 共1兲 共2兲

N M ␳one electron共x兲 = 兺i=1 ␳i共x兲 = 兩⌿兩2, the first be a factorization of the total density ␳共x兲 = 兺n=1 equality from the equivalence of the electrons, and provide the square root of each one of the M shell structure contributions ␳i共x兲 = 兩bi␾i兩2 to the total density.

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Quadratic forms and electronic structure

C. The energy calculation

We rewrite the usual expression for the total nonrelativistic electronic energy operator or ˆ in correspondence with our formal definition of configuration space 共an atomic Hamiltonian H 0 electron structure is used in the following to simplify the notation, otherwise a summation ⌺A and the relative distances xaA should be used in the electron-nucleus potential energy兲. We use the ¯ b其 to select, from the wave function ⌿NKKW above-defined per electron geometric notation 兵␻b , ␻ KKW and its conjugate ⌿N , the corresponding contributions. A per electron operator is an effective one-body operator. We consider first the per electron “a” kinetic energy operator 2 ¯a /2me兲␻ − ␻a共ប2ⵜ共x a兲

and the electron a to nucleus A 共considered at the origin of coordinates兲 potential energy operator ¯ a. The sum of the contributions of these two terms will be called “core” energy in −␻a共ZAe2 / 兩xa 兩 兲␻ the following. Coul between the electrons: a half of the sum over Second, the pairwise Coulomb interaction Ve−e, all a, of the for electron “a” in the pair a , b, electron-electron potential energy, we have Coul Ve−e =

冉

冕

冊

1 N e2 ˆ ¯a ¯b ␻ ␻ d⍀ ␻ ␻ 兺 a N−a 兺 b 2 a=1 兩xab兩 b⫽a

ˆ integration for all b ⫽ a each term has been 共notice that after performing the indicated 兰d⍀ N−a reduced to a per electron operator兲. Then

冋

ˆ = 兺N ␻ − H 0 a=1 a

2 ប2ⵜ共x a兲

2me

冕

N

冉

e2 Z Ae 2 1 ˆ ¯b + − d⍀N−a 兺 ␻b ␻ 兩xab兩 兩xa兩 2 b⫽a

冊册

¯ a. ␻

共3兲

Correspondingly, the wave function ⌿NKKW is 共first in a per electron n 傺 N basis and second in a per orbital i 傺 M description, using the geometric operators ␣i per auxiliary basis function ␾i 共SO兲 with weight bi, obeying ␣i␣ j = −␣ j␣i, j ⫽ i, and the projection operators ¯␣i␣ j = ␦ij兲: N

⌿NKKW = 兺 共␻␺兲n ,

共4兲

n=1

冋兺

册

M艌N

where 共␻␺兲n = ␻n

bi␣i␾i共xn兲 ,

i=1

共5兲

N

and

⌿NKKW = 兺 共¯␺␼兲n ,

共6兲

n=1

冋兺

册

M艌N

where 共¯␺␼兲n =

b*i ¯␣i␾*i 共xn兲 ␼n ,

i=1

共7兲

with normalization N=

冕

N

⌿NKKW⌿NKKWd⍀N ⬟ 兺

c=1

冕

¯ c⌿NKKWd⍀c其, 兵⌿NKKW␻c␻

共8兲

and, when written in terms of the auxiliary spin-orbitals ␾i,

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J. Keller and P. Weinberger

N=N

冕

M艌N

兺 i=1

¯␺␺d⍀ = N

冕

M艌N

␾*i b*i ␣i␣ibi␾id⍀

=N

兺 i=1

兩bi兩

2

冕

M艌N

兩␾i兩 d⍀ ⬟ N 2

兺 i=1

兩bi兩2具i兩i典.

共9兲

In this paper we assume equal number of spin up and spin down electrons, the spin restricted case. The second line in 共8兲, from the definition of 兰d⍀N in Sec. II A is an identity. In 共9兲 we use the orthonormality generated by ␣i␣ j = ␦ij. Note that a double set of Grassmann 共anticommuting兲 numbers 兵␻a ; ␣i其 has been introduced, this has an analytical analogue in the determinants of the ab initio methods where either the exchange of columns or of rows change the sign of the determinant. Formally the wave function ⌿ could be represented by a rectangular M ⫻ N matrix with entries ␣i␻a. The second line in 共4兲 ¯ ⌿. Exchange terms will arise from 共4兲 would correspond to the 共real positive number兲 trace of ⌿ when used in 共3兲. The ␾i’s are defined to form an orthonormal set of spin-orbitals 具i 兩 j典 ⬅ 兰␾*i ␾ jd⍀ = ␦ij␦ss j. The ␦ss j ensures that the spin of i and j are the same, this is superfluous here i i but will be used in the following. We introduce the normalization 兺i 兩 bi兩2 = 1, then the one electron density ␳1共x兲 = ¯␺共x兲␺共x兲 = 兺i 兩 bi兩2␾*i 共x兲␾i共x兲. ˆ for electron n: Hcore共x兲 + Hinteraction共x兲 where the second term is an In 共3兲 the Hamiltonian H 0 effective local one-electron operator, even if the electron repulsion, being dependent on the interelectron distance, is a two-electron 共i for n, j for m兲 operator. The resulting potential is the same for all components of ␺. D. Core energy N ˆ core共c兲␻ ¯ c, then Rewrite the first two terms in 共3兲 as 兺c=1 ␻ cH N

Ecore = 兺

c=1

冕

N

¯ 兵␻ H ˆ core共c兲␻ ¯ c其⌿d⍀c = 兺 ⌿ c

c=1

冕兺 d⬘

ˆ core共c兲␻ ¯ d⬘¯␺d⬘共xd⬘兲兵␻cH ¯ c其 ⫻ 兺 ␻d␺d共xd兲d⍀c , ␻ d

共10兲 and from 共5兲 and 共7兲, orthonomality and equivalence: Ecore = N

冕

¯␺ 共x 兲H ˆ core共1兲␺ 共x 兲d⍀ = N 1 1 1 1 1

冕兺

ˆ core␾ d⍀. 兩bi兩2␾*i H i

共11兲

i

The above-mentioned 兵␻c , ␼c其 and the 兵␣i , ␣i其 have selected the sum of the diagonal elements in 共10兲. Note that the shell structure cannot be avoided.8,9 E. Electron-electron interaction energy

For the electron-electron interaction 共e-e兲 Ee-e =

冕

⌿NKKW兺 a ␻a ⫻

冋

N

1 兺 2 b⫽a

冕冉

␻b

冊

册

e2 ¯ b d⍀b ␻ ¯ a⌿NKKWd⍀a . ␻ 兩xab兩

共12兲

Here, from the equivalence of the N electrons, we have N equal pairwise 兵1 Û 2其 contributions which consider all spin-orbitals. Using the expansion of the ␺, we obtain Ee-e =

N 2

冕冕 兺 i

␾*i 共1兲b*i ¯␣i 兺 ␾*i 共2兲¯␣ j j

e2 ⫻ 兺 ␾i共2兲␣k 兺 ␾i共1兲␣ld⍀1d⍀2 . 兩x12兩 l k

共13兲

Considering the property ¯␣i␣ j = ␦ij there are three types of e-e terms: 共I兲 j = k and i = l which gives

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Quadratic forms and electronic structure

N 2

冕 再兺 冕 冋 i

=

兺 兩b j兩2兩␾ j共2兲兩2 j⫽i

冎

册

e2 1共x2兲d⍀2 ⫻ 兩bi兩2兩␾i共1兲兩2 1共x1兲d⍀1 兩x12兩

N N 兩b j兩2兩bi兩2共ij,ij兲 = EI , 兺 兺 2 i j⫽i 2

共14兲

where we have introduced the notation 共ij , ij兲 for the integrals, as they are “Coulomb integrals” in the accepted electronic structure calculation language. Also the formal local unit factor 1共x兲 = ¯␺共x兲␺共x兲 / ␳1共x兲 will be fundamental to perform the variational procedure in the following to find the effective equations for the ␺共x兲. Note that the 1共x兲 appear twice in this term, this will result in a factor 2 in the variational wave equation. 共II兲 j = l ⫽ i and i = k 共the ␦ss j from spin orthonormality, and one change of sign from the i interchange ␣i␣ j = −␣ j␣i is needed!兲 −

N 兺 ␦s j兩b j兩2兩bi兩2 ⫻ 2 i,j⫽i si =−

冕冕 冋

␾*j 共1兲␾*i 共2兲

册

e2 ␾i共1兲␾ j共2兲 d⍀21共x1兲d⍀1 兩x12兩

N N ␦ssij兩b j兩2兩bi兩2共ji,ij兲 = EII , 兺 2 i,j⫽i 2

共15兲

where we have used the notation ␦ss j共ji , ij兲 for the integrals, as they are “exchange integrals” i corresponding to the 关i , j兴 pair of spin-orbitals in the accepted electronic structure calculation language; the ␦ss j requires si = s j and ensures that the product ␾*j 共1兲␾i共1兲 of the SO, with spin si and i s j, respectively, is not null. We use again the formal local unit factor 1共x兲. 共III兲 Null terms, all others, where 共i ⫽ l and i ⫽ k兲 or 共j ⫽ l and j ⫽ k兲. EI and EII also contribute to the formal interpretation of the Pauli exclusion principle: first a given electron is not interacting with itself and, second, there is an “exchange” term for fermions, where from ␣i␣ j = −␣ j␣i a negative sign appears. ¯ III. VARIATION OF THE TOTAL ENERGY WITH RESPECT TO THE ␺

Here we consider the total energy in terms of the ␺ as noted earlier. From the normalization 共9兲 the per electron density is ␳one-electron共x兲 = ¯␺共x兲␺共x兲 and the total density N␳one-electron共x兲. We write the total energy Etotal = Ecore +

N e-e 共E + EIIe-e兲. 2 I

共16兲

For the core term Ecore/N =

冕

the variation with respect to ¯␺1共x1兲 gives

¯␺ 共x 兲H ˆ core共1兲␺ 共x 兲d⍀ , 1 1 1 1 1

再

ˆ core共1兲␺ 共x 兲 = − H 1 1

2 ប2ⵜ共1兲

2me

−

冎

Z Ae 2 ␺1共x1兲. 兩x1兩

For the variation of EIe-e / 2 with respect to ¯␺1共x1兲 we obtain, defining VI共x1兲 = 兺 i

再

兩bi兩2兩␾i共x1兲兩2 ⫻ ␳1共x1兲

冕 冋兺

j⫽i

共17兲

兩b j兩2兩␾ j共2兲兩2

e2 兩x12兩

共18兲

册冎

d⍀2 ,

共19兲

and, considering that the factor 1共xk兲 appears twice in 共14兲, the repulsive electron-electron term

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2 21 VI共x1兲¯␺1共x1兲 = VI共x1兲¯␺1共x1兲.

共20兲

Finally for the variation of EIIe-e / 2 with respect to ¯␺1共x1兲 we obtain 共assuming here, for simplicity not as a restriction in the method, to avoid introducing summations over spin coordinates, equal number of spins “up” and “down”兲, defining

VII共x1兲 = −

兺 i,j⫽i

␦ssij兩b j兩2兩bi兩2 2␳1共x1兲

⫻

冕冋

␾*j 共1兲␾*i 共2兲

册

e2 ␾i共1兲␾ j共2兲 d⍀2 , 兩x12兩

共21兲

the “attractive-like functions interchange” electron-electron term VII共x1兲¯␺1共x1兲.

共22兲

A. The KKW auxiliary equations

The variational procedure has been carried with respect to the ¯␺’s, to obtain the formal equation which describes any electron n in the system 共reminder ␺ is a vectorial sum of functions ˆ ⌿ d⍀ from 共11兲 to 共15兲 as in 共16兲, and performed ␾i兲. We collected the different terms of 兰⌿N† H 0 N the variation with respect to ¯␺共x兲, as shown in the previous section. We obtain all together an KKW ˆ eigenvalue equation effective Hamiltonian H KKW ˆ H ␺共x兲 = ␮␺共x兲,

再

冎

ប2ⵜ2 Ze2 KKW ˆ H + VI共x兲 + VII共x兲 . ⬅ − − 2me 兩x兩

共23兲

This is not yet a practical equation. Use now the expansion of the ␺ in terms of the ␾ for a further reduction. Write 共23兲 as M艌N M艌N KKW ˆ H 兺i=1 bi␣i␾i共xn兲 = ␮ 兺i=1 bi␣i␾i共xn兲,

共24兲

apply on both sides the projector ␣i to obtain the practical equations for the set of auxiliary orthonormal functions ␾i 共from ¯␣i␺ = bi␾i兲, KKW ˆ H ␾ i = ␧ i␾ i .

共25兲

Finally we obtain, by left multiplication with ¯␺ of 共23兲, integration and the normalization 具␺ 兩 ␺典 = 1, 具␾i 兩 ␾i典 = 1, a relation between the ␮ and the ␧i’s given by ␮ = 兺i␧i 兩 bi兩2 = ¯␧, that is: ␮ in 共23兲 is the weighted average eigenvalue.

B. A more familiar and practical form of the auxiliary wave equations

We can rewrite the electron-electron interaction energy in a computationally more practical form 共related to the ab initio methods.1兲 Consider Ee-e =

N 兵EI + EII其, 2

and rewrite as

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083505-9

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Quadratic forms and electronic structure

Þ

N 2

冕

¯␺共x 兲兵E 共x 兲 + E 共x 兲其␺共x 兲d⍀ , 1 Coul 1 XC 1 1 1

obtained if, in the above-presented definitions, the “self-Coulomb” integrals 兩bi兩2共ii , ii兲 are added to remove the condition j ⫽ i, to EI. The “self-exchange” integrals −兩bi兩2共ii , ii兲 are added to remove the condition j ⫽ i, to EII. 关The EII is now as in the formally equivalent considerations of the first part of the Slater 共1951兲 idea,13 which here is no longer an approximation.兴 Then

再

冎

ប2ⵜ2 Ze2 KKW ˆ H + VCoul共x兲 + Vxc共x兲 , ⬅ − − 2me 兩x兩

VCoul共x1兲 = N

Vxc共x1兲 = −

冕冋

N 兺 ␦s j兩b j兩2兩bi兩2 ⫻ ␳共x兲 i,j si

␳共2兲

冕冋

␾*j 共1兲␾*i 共2兲

By substitution of 共26兲 in 共25兲

再

−

册

e2 d⍀2 , 兩x12兩

册

e2 ␾i共1兲␾ j共2兲 d⍀2 . 兩x12兩

冎

ប2ⵜ2 Ze2 + VCoul共x兲 + Vxc共x兲 ␾i共x兲 = ␧i␾i共x兲. − 2me 兩x兩

共26兲

共27兲

KKW ˆ is the same for all ␾i’s. Reminder H The total energy is obtained by direct integration of the set of equations 共27兲 multiplied on the left by 兺i 兩 bi兩2␾*i and comparing with 共16兲:

E关⌿兴 = 兺 兩bi兩2␧i − i

ECoul − 2

冕

VXC共x兲␳共x兲d⍀ + EXC

共28兲

共we must remember that ECoul and EXC include the “self-Coulomb” and the “self-exchange,” respectively兲. With additional variational constants bij defined through bi = 冑共1 − 兺 j⬎Nbij兲 / N for i 艋 N and b j = 冑兺i艋Nbij兲 / N j ⬎ N, we obtain a 共twice兲 variational procedure to obtain a set 兵bij其 and the ␾i. A secular determinant can be constructed and solved. If the basis 兵␾i其 is large enough a timedependent formulation with ⌬V共t兲 can be constructed where bi Þ bi共t兲 and the bij Þ bij共t兲 describe induced transitions. Finally the basic definition for the total energy E = E关¯␺共x兲␺共x兲兴 of N equivalent carriers 共electrons兲, can be formally written in terms of the density N¯␺共x兲␺共x兲 defining

Ecore = N

冕 冋兺 i

ˆ core␾ 共x 兲 兩bi兩2␾*i 共x1兲H i 1 ␳1共x1兲

册

⫻ ¯␺共x1兲␺共x1兲d⍀1 = N

冕

␧core共x1兲¯␺共x1兲␺共x1兲d⍀1 . 共29兲

E=N

冕

兵␧core共x1兲 + ␧inter共x1兲其¯␺共x兲␺共x兲d⍀1 ,

共30兲

and, considering E = N 兰 ␧¯␺共x兲␺共x兲d⍀1, we define ␧core共x1兲 + ␧inter共x1兲 = ␧.

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083505-10

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IV. THE PROPERTIES OF THE WAVE FUNCTION: CONCLUSIONS

The KKW wave function describes a system of N electrons with an expansion based on M 艌 N auxiliary, mutually orthogonal functions, solutions of the same “KKW” Hamiltonian. The complete wave function, then the one density, is directly optimized and not each auxiliary function at a time. From the many possibilities to construct the ␺ it is useful to choose the one presented here because the eigenvalues ␧i, by construction the rate of change of the total energy with respect to occupation of the “i” spin-orbital, are an approximation to the removal energy of one electron, from that ␾i, in the system. Also because the SO functions, being related to the response function of the system, have become observables. • The KKW method offers a complete variational solution to the problem of finding the wave function and simultaneously the energy of the ground state of the N electron system. • To find the limit for the ground state energy of the system the number of auxiliary functions to be considered is M ⬎ N. The practical approach to this calculation will be twice variational, first in the sense that the auxiliary functions are found from the variation of a functional, and second because the weight of the contributions of the set of M ⬎ N functions requires a set of variational parameters to be found. • The new method systematically includes the kinetic and the potential energy for all the M
N auxiliary functions. It has the advantage that all spin-orbitals are simultaneously optimized. An alternative would be to obtain M ⬘ ⬎ N functions from the results for M = N and, in a second step, the best possible bij for the M ⬘ case. • The calculations can include the use in the Hamiltonian of selective pairwise interaction terms ¯ b␻ ¯ a, 共⌬Ve-e兲ab = ␻a␻b共⌬Vmn兲␻ allowing the description of pair-correlations of any origin: magnetic, or electron-phonon ¯ a. or from any other indirect type. Also specific one spin-orbital cases 共⌬V兲a = ␻a共⌬Va兲␻ This being a further advantage of the geometric notation. • The self-consistent solution requires, in general, numerical solutions. Nevertheless all analytic and computational methodologies in usual practice can be used without major changes. Comparison with previous methodologies: Conclusions. In practice several features made the, now known as the Hartree-Fock method 共HF2,3兲, the reference for atomic, molecular, and ground state condensed matter calculations: it is a formally correct variational procedure based on the use of a determinant of auxiliary functions for which a differential equation is deduced within the method. It includes 共through the basic properties of the Slater determinant兲 the Pauli exclusion principle as well as the indistinguishability of equivalent fermions. HF is also a suitable starting calculation for establishing the procedure known as configuration interaction 共CI兲, introducing a set of variational constants, when the spin-orbitals resulting from the HF calculation are used to construct a formally complete wave function as a sum of mutually orthogonal Slater determinants. The standard definition of “exchange energy” and of “correlation energy” is given in relation to the HF+ CI procedures. In the determinants no two-electron-relative-coordinates functions are used, neither in these approaches nor in the one described earlier. Another widely used methodology, based on the density functional theory 共DFT兲 a correct formal procedure itself,14,15 can be considered as related to HF+ CI as the standard calculations include local density functionals for the exchange and correlation energies and potentials. The methodology developed here shares both all the favorable features of the HF+ CI method and the advantages of the DFT procedure. This suggests that there should be a relation among all three procedures HF+ CI, DFT, and KKW. • The KKW method 共M = N兲 requires the self-consistent solution of N differential equations,

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083505-11

Quadratic forms and electronic structure

J. Math. Phys. 47, 083505 共2006兲

as in DFT, unlike the HF which is an N ⫻ N formulation. Compared to the ab initio techniques, our methodology employs a simpler set of equations. • If, as in the HF method, using M = N, each auxiliary function were to be optimized separately one at a time in a self-consistent set, from our expressions 共16兲 for the total energy, we go back to the HF procedure and results. Our method has the advantage that the complete set is optimized simultaneously to search for the lowest total energy. • The KKW method in the minimum auxiliary functions procedure is equivalent to the Slater 共1951兲 average exchange proposal 共with the so-called “exact” exchange兲 and in fact transforms this Slater “approximation” into a nonapproximate procedure where all auxiliary functions are optimized simultaneously. This is independent of the DFT approximation for EXC. • The formal structure of the new method enriches the methodology of the DFT by showing that the Slater exchange and the Kohn-Sham procedures are formally integrated to the general scheme and also because an N-electron wave function ⌿, which is required in the theorems of the formal DFT theory, is constructed and used within the formalism. The KKW approach also shows, from the different analysis presented here, DFT both as an ab initio and as a first principles method. This is an important formal contribution of our study. In the following we quote standard use of the so called ab initio and DFT methods, see, for example Refs. 1, 16, and 17. Computationally there is no need to consider a self-Coulomb energy. As all electrons have equivalent descriptions, with the same density per electron, there is an exact resultant factor 共共N − 1兲 / N兲 in the electron-electron Coulomb interaction. It is basic for the calculation of the hydrogen atom. It is dominant in the calculation of the helium atom where each of the two electrons interact with the other but not with itself, and progressively less important for larger systems where the interchange part of the interaction grows. Standard DFT programs18,19 for atomic electronic structure calculations are easily modified, an 共N − 1兲 / N factor in the Coulomb potential, replacement of the exchange-correlation and total energy subroutines, to solve the KKW equations 共27兲. As a numerical test the relativistic program “David”19 was adapted, the H atom calculation used as a first check of the numerical procedures, the He atom with M = N, the Hartree-Fock limit for the Coulomb interaction, and the He atom with 共1s , 2s , 2p兲 for the M ⬎ N case, beryllium and krypton in the M = N as further tests. All results are acceptable in the limit of the corresponding approximations 共Hartree atomic units兲: Atom H He Be Ne Kr

Etotal −0.5 −2.886 −14.603 −128.87 −2799.49

EHF −0.5 −2.864 −14.573 −128.55 −2796.72

EDFT ¯ −2.867 −14.592 −128.62 −2787.80

Eexp −0.5 −2.90 −14.673 −128.94 ¯

In our analysis we have gone beyond complex algebra and calculus, in fact we have gone to the more general domain of the Grassmann-Clifford algebra and analysis. ACKNOWLEDGMENTS

J.K. wishes to gratefully acknowledge the hospitality and fundamental discussions with the group of Professor Peter Weinberger, and also the technical assistance of Irma Vigil de Aragón. A preliminary form of this methodology was presented by J.K. and Alejandro Keller at the Seventh International Conference on Clifford Algebras, 2005 Toulouse. This project was additionally supported by the UNAM Program PAPIIT No. IN113102-3. A. Szabo and N. S. Ostlund, Modern Quantum Chemistry 共Dover, New York, 1996兲. D. R. Hartree, Proc. Cambridge Philos. Soc. 24, 89 共1928兲; 24, 111 共1928兲; 24, 426 共1928兲. 3 V. A. Fock, Z. f. Phys. 39, 226 共1926兲; 61, 126 共1930兲. 4 J. Itatani et al., Nature 共London兲 432, 867 共2004兲. 1 2

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P. Jörgensen, Annu. Rev. Phys. Chem. 26, 359 共1975兲. M. A. L. Marques and E. K. U. Gross, Annu. Rev. Phys. Chem. 55, 427 共2004兲. 7 A. Worth and L. S. Cederbaum, Annu. Rev. Phys. Chem. 55, 127 共2004兲. 8 J. Keller, J. A. Flores, and A. Keller, Folia Chim. Theor. Lat. 18, 175 共1990兲. 9 J. A. Flores and J. Keller, Phys. Rev. A 45, 6259 共1992兲. 10 J. Keller, The Theory of the Electron; A Theory of Matter from START, Foundations of Physics Series Vol. 117 共Kluwer Academic, Dordrecht, 2001兲. 11 J. Keller and P. Weinberger, Adv. in Appl. Clifford Algebras 12, 39 共2002兲. www.clifford-algebras.org 12 J. Keller, in What is the Electron?, Modern Structures, Theories, Hypotheses, edited by V. Simulik 共Apeiron, Montreal, 2005兲, pp. 1–28. 13 J. C. Slater, Phys. Rev. 81, 385 共1951兲. 14 P. Hohenberg and W. Kohn, Phys. Rev. 136, 864 共1964兲. 15 W. Kohn and L. J. Sham, Phys. Rev. 140, 1133A 共1965兲. 16 S. P. McGlynn, L. G. Vanquickenborne, M. Kinoshita, and D. G. Carroll, Introduction to Applied Quantum Chemistry 共Holt, Rinehart and Winston, New York, 1972兲. 17 W. Koch and M. C. Holthausen, A Chemist’s Guide to Density Functional Theory, 2nd ed. 共Wiley-VCH, Weinheim, 2002兲. 18 D. A. Liberman, Comput. Phys. Commun. 2, 107 共1971兲. 19 D. A. Liberman, Comput. Phys. Commun. 32, 63 共1984兲. 5 6

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