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Anticommuting space : an alternative formulation of the wavefunction antisymmetry description
Kleber C. Mundim
To cite this version:
Kleber C. Mundim. Anticommuting space : an alternative formulation of the wavefunction antisym- metry description. Journal de Physique, 1989, 50 (1), pp.11-17. �10.1051/jphys:0198900500101100�.
�jpa-00210895�
Anticommuting space : an alternative formulation of the wavefunction antisymmetry description
Kleber C. Mundim (*)
Centro Brasileiro de Pasquisas Fisicas, R. Xavier Sigaud 150, 22290 Rio de Janeiro, RJ, Brasil
(Reçu le 21 avril 1988, accepté sous forme définitive le 23 août 1988)
Résumé. 2014 Nous montrons le rôle important joué par les algèbres antisymétriques en chimie quantique. Nous utilisons les propriétés des espaces multilinéaires alternés pour imposer l’antisymétrie de la fonction d’onde et calculons certains invariants moléculaires.
Abstract. 2014 We show the important role of the anticommuting algebra in quantum chemistry. We
use the properties of the Multilinear Alternating Space in order to obtain the antisymmetry of the
wave function and we calculate some molecular invariants.
Classification
Physics Abstracts
31.15
1. Introduction.
As is well known, the antisymmetry of the wavefunction t/1 and the Pauli principle were
formulated by defining the functions tb as linear combinations of Slater determinants [1]. The
second quantization formalism [2] describes them in a certain way through the anticommuting operators. A different formulation could be to use the properties of an anticommuting
variable space. The Grassmann anticommuting algebra - which has been known for a long
time by mathematicians - was used by Berezin and Marinov [3-4] to develop and to explore
an application of the functional generator method in the second quantization theory.
Schwinger [5], Matthews and Salam [6] used the anticommuting c-Numbers in studies of fermionic systems. Since then, due to the operational adequacy of Grassmann algebra to
describe systems of fermions and supersymmetry, many authors have been working on this subject [7-10].
We shall present an alternative formulation to the description of the antisymmetric properties of the wavefunction and the Pauli principle, and we show that the antisymmetry of
the Slater determinant becomes a particular case of this formulation. It is worth to underline that these known ideas appear, through this approach, in a much simpler and straightforward
way. We define the state vectors starting from the generators of a multilinear altemating
space g, which is introduced as an N-linear alternating application of the Hilbert space Je, A:JC x JC x ... x X --+ 9.
(*) On leave of absence from the Dep. de Fisica, UFBA, 40000 Salvador, 40000 Salvador-BA, Brasil.
Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphys:0198900500101100
12
2. Multilinear alternating application.
In this section we present some properties of a linear alternating space [11] in order to make
the next steps and definitions clear.
Since El, ..., E,, U are vectorial spaces, an application ;
is named r-linear if it is linear in each variable separately, el E El, ..., ek and q e Ek, ..., e, E Er and a e 3t, so that
and
We denote as £(El, ..., E,, ; U ) the group of r-linear applications.
An r-linear application f : E x ... x E -+ 91 is called alternating when
whenever the sequence (e 1, - - -, e k, ..., e,) has repetitions. In other words
where §1 , ..., e,, e E E.
In order that E C,(E, CU,) be alternating, it is necessary and sufficient that is antisymmet- ric, that is :
We observe that the r-multilinear alternating maps fulfil conditions completely analogous
to those of determinants. Thus, the multilinear alternating maps can be understood as a
generalization of determinants.
Proposition : f : E x ... x E - U is a r-linear alternating application. Since we have an
ordered basis (y,, ..., y.) at E, for each subgroup J = {j1 ’-- j2 « - - - «- j,} c lm = {l, 2, ... , m ) , and
are vectors of E, we have
where « J is a r x r submatrix of the r x m matrix built from the vectors of coefficients
e,. The maximum number of different elements uj E ’11 is given by combining r generators of
a m-dimensional space,
3. Spin-orbitals in the Hilbert-Grassmann space.
In this section we define the spin orbitals as products among generator elements of the 9- space and we show that this definition is a generalization of the spin orbital definition as a
linear combination of Slater determinants.
For this purpose we shall define the vectorial space E which we have mentioned in section 2.1, equivalent to the Hilbert space Je. So an N-linear altemating map in the space JC will be [12] :
and we call the space 9 a Hilbert-Grassmann space.
So, we can define the state vectors of an N-electron system
Using multilinear alternating map properties (Sect. 2.1) we can easily show that the vector in equation (11) obeys the Pauli principle, that is
and
if ej = ek ; k * j and ej A ek is a tensor product.
As is known, the N-multilinear alternating maps can be understood as a generalization of
determinants. So, when we use an appropriate basis, yi, y2, ..., YN, we can show that our N- linear map on the Hilbert space Je gives the following equation :
where the matrices ai are the N x N matrices defined by the vector components in the Hilbert space.
As an example, let us give the state vector of a two-electron system
where
ç k are the one-electron basis generators, which have been defined as a linear combination of a
unitary base and I/Il is a complex function of particle k. Thus
JOURNAL DE PHYSIQUE. - T. 50, N- 1, JANVIER 1989
14
Since y k A Y k = 0, the first and last terms are zero ; the second and third terms are related by
l’ j A l’ k = - l’ k A l’ j’ j =1= k. Thus
where the term in parentheses is the Slater determinants of a two-electron system.
4. Applications.
In order to show the important role of exterior algebra for quantum-chemical models of the many-electron system, we calculate some molecular invariants.
4.1 HARTREE-FOCK ELECTRONIC GROUND STATE ENERGY. - We can easily show that the generators ek e 9 and ekr = g fulfil the same properties as the conventional annihilation and creation operators ak and at
As in a conventional case, an annihilation element ek removes or annihilates an electron in the k-th orbital
and
We have then the equivalence
The annihilation element is thus defined in the conjugated Igt space.
Due to the definition of a multilinear alternating space generator, we have
As we have stated, the annihilation elements ek and the annihilation operators ak share the same anticommutation properties. From equations (22) and (23), we have
which has the immediate consequence that
It means that one electron in the k-th orbital can be annihilated only once, as the second
annihilation element ek finds a vacancy. It is precisely the Pauli principle.
In the general case, we have
The adjoint of the element ek is defined by
Now, we shall apply what precedes to estimate the Hartree-Fock ground state energy of a
closed shell N-electron system
We define the one-electron operators through the creation and annihilation elements :
For the one-electron operator we have
where we used equation (26) in order to move ej to the right side,
Let us remark that (1 ... N Il ... N) = 1 and the second term is zero because
Thus
Where hj is the expectation value of the one-electron core hamiltonian corresponding to the
molecular orbital. For the two-electron interaction term we define the following operator,
The contribution to the total energy due to the repulsion between electrons is hence written
as
Using the same arguments as for the one-electron case, we have
16
As in the first term j = m, i = k and in the second term j = k, i = m
Integrating over spin variables in the terms hk, g; and gji we have the terms Hk,
Jjj, K;
and
where Jij and Kij are the Coulomb integrals and exchange integrals respectively.
The total electronic energy is then ,
which is the well known expression of the Hartree-Fock total electronic energy for a closed shell ground state.
4.2 N-ORDER DENSITY FUNCTIONAL. - In this section we introduce the n-order density
functional from the inner product between the éléments e K and ei E Je t. We show that the definition of inner product in a 2-linear altemating space may be used in order to obtain
an expression linking the bond index 1AB and the correlation between the atomic charge
fluctuations in a molecule. Pitanga and Giambiagi found, recently, an analogous expression by using a statistical interpretation of 1AB [13].
We introduce the first-order density matrix (pi ) by :
where o, is the spin part of the electron i.
From equation (44) we have,
and summing up over spin ui and o-j,
Which is a consequence of the Schwartz inequality.
Let us consider the determinant of the U)2 matrix ;
Performing the summation over all orbitals i E A, j e B and all spins ui and uj, we have
or
where (qA) is the Mulliken charge of atom A and 1 AB are the bond indices between atoms A
and B, or in an orthogonal basis the Wiberg indices,
The right side of equation (49) expresses the correlation between the atomic charges of
atoms A and B.
In short, we see that the antisymmetry of the wavefunction built from Slater determinants
belongs to a general mathematical framework. This allows us to calculate important
molecular invariants synthetically.
References
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[4] BEREZIN, F. A. and MARINOV, M. S., Ann. Phys. 104 (1997) 336.
[5] SCHWINGER, J., Phys. Rev. 82 (1951) 914.
[6] MATTHEWS, P. T. and SALAM, A., Nvo. Cim. 2 (1955) 120.
[7] FINKELSTEIN, R. and VILLASANTE, M., Phys. Rev. D 33 (1986) 1666.
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[11] LANG, S., « Algebra », 2nd ed. Addison-Wesley (London) 1984.
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