Slater-determinant

Slater determinants have a number of interesting properties. First, note that every electron appears in every spin orbital somewhere in the expansion. This is a manifestation of the indistinguishability of quantum particles (which is violated in the Hartree-product wave functions). A more subtle feature is so-called quantum mechanical exchange. Consider the energy of interelectronic repulsion for the wave function of Eq. (4.43). We evaluate this as 

Equation (4.49) indicates that for this wave function the classical Coulomb repulsion between the electron clouds in orbitals a and b is reduced by Kab, where the definition of this integral may be inferred from comparing the third equality to the fourth. This fascinating consequence of the Pauli principle reflects the reduced probability of finding two electrons of the same spin close to one another - a so-called Fermi hole is said to surround each electron. 

Note that this property is a correlation effect unique to electrons of the same spin. If we consider the contrasting Slater determinantal wave function formed from different spins 

Note that e disappearance of the exchange correlation derives from the orthogonality of the 0 and p spin functions, which causes the second integral in the second equality to be zero when integrated over either spin coordinate. 

Now let us select an ordered tuple of N subscripts referring to spin-orbitals K= ki k2 . .. I n. The (V-tuple of spin-orbitals defines uniquely the Slater determinant of N electrons as a functional determinant 

It can be proven [31] that all possible Slater determinants of N particles constructed from a complete system of orthonormalized spin-orbitals 4 k form a complete basis in the space of normalized antisymmetric (satisfying the Pauli principle) functions, of N electrons i.e. for any antisymmetric and normalizable (K one can find expansion amplitudes so that  

Thus the basis of Slater determinants can be used as a basis in a linear variational method eq. (1.42) when the Hamiltonian dependent or acting on coordinates of N electrons is to be studied. The problem with this theorem is that for most known choices of the basis of spin-orbitals used for constructing the Slater determinants of eq. (1.137) the series in eq. (1.138) is very slow convergent. We shall address this problem later. 

The general setting of the electronic structure description given above refers to a complete (and thus infinite) basis set of one-electron functions (spin-orbitals) (f n T (x). In order to acquire the practically feasible expansions of the wave functions, an additional assumption is made, which is that the orbitals entering eq. (1.136) are taken from a finite set of functions somehow related to the molecular problem under consideration. The most widespread approximation of that sort is to use the atomic orbitals (AO).17 This approximation states that with every problem of molecular electronic structure one can naturally relate a set of functions y/((r). // = M N -atomic orbitals (AOs) centered at the nuclei forming the system. The orthogonality in general does not take place for these functions and the set y/ is characterized 

17It may not seem mandatory now, with the advent of plane wave basis sets. However, to give a better description, these latter are variously augmented to reproduce the behavior of electrons in the vicinity of nuclei. For more detailed description see [35] and reference therein. 

Since bosons require a wave function symmetric with respect to interchange, there is no restriction on the number of bosons in a given state. 

Slater pointed out in 1929 that a determinant of the form (10.40) satisfies the antisymmetry requirement for a many-electron atom. A determinant like (10.40) is called a Slater determinant. All the elements in a given column of a Slater determinant involve the same spin-orbital, whereas elements in the same row all involve the same electron. (Since interchanging rows and columns does not affect the value of a determinant, we could write the Slater determinant in another, equivalent form.) 

Consider how the zeroth-order helium wave functions that we found previously can be written as Slater determinants. For the ground-state configuration (Is) we have the spin-orbitals Isa and ls/3, giving the Slater determinant 

Comparison with (10.27) to (10.30) shows that the ls2s zeroth-order wave functions are related to these four Slater determinants as follows  

Next consider some notations used for Slater determinants. Instead of writing a and j8 for spin functions, one often puts a bar over the spatial function to indicate the spin function p, and a spatial function without a bar implies the spin factor a. With this notation, (10.40) is written as 

While idly dreaming over these equations (theoretieians eall it working ) we might happen to notiee that the linear eombination (9-24b) we have seleeted for ground-state helium is the same as the expansion of a 2 x 2 determinant 

Manipulating linear eombinations for Li, one soon diseovers that the only one that satisfies the indistinguishability prineiple for eleetrons is the expansion of a Slater determinant 

Moreover, there are 2 terms in the expansion of the Slater determinant for He but there are 6 terms for Li. Looking at beryllium, we find 24 terms. This is the beginning of the faetorial series 

When we square the wave funetion, we expeet to find a probability P = 1 over all spaee, so the n terms in the expanded determinant must be multiplied by the faetor 1 /voj to obtain the determinantal wave funetion normalized to 1. 

The top row of the Slater determinant shows no preferenee for any spinorbital j) over any other the eleetron may be in any one of them with equal 

The simplest possible LCAO building block for the ground state of dihydrogen is 

This simple wavefunction is antisymmetric to the exchange of electron names, and treats both space and spin. 

Determinants have the useful property that they change sign when we interchange two columns, and this is equivalent to interchanging the names of the two electrons  

Such a determinantal wavefunction is called a Slater determinant, after Slater (1929), and you should appreciate that a 

Slater determinant = smallest logical building block for electronic 

There should be something in the theory which assures us that, if we renumber the electrons, no theoretical prediction will change. The postulate of the antisymmetric character of the wave function with respect to the exchange of the coordinates of any two electrons, certainly ensures this (Chapter 1, p. 28). The solution of the Schrodinger equation for a given stationary state of interest should be sought amongst such functions. 

A Slater determinant is a function of the coordinates of N electrons, which automatically belongs to Cl  

A Slater determinant carries two important attributes of the exact wave function  

Consider some points about the Pauli exclusion principle, which we restate as follows In a system of identical fermions, no two particles can occupy the same state. If we have a system of n interacting particles (for example, an atom), there is a single wave function (involving 4n variables) for the entire system. Because of the interactions between the particles, the wave function cannot be written as the product of wave functions of the individual particles. Hence, strictly speaking, we cannot talk of the states of individual particles, only the state of the whole system. If, however, the interactions between the particles are not too large, then as an initial approximation we can neglect them and write the zeroth-order wave function of the system as a product of wave functions of the individual particles. In this zeroth-order wave function, no two fermions can have the same wave function (state). 

In 1925, Einstein showed that in an ideal gas of noninteracting bosons, there is a very low temperature (called the condensation temperature) above which the fraction/of bosons in the ground state is negligible but below which/becomes appreciable and goes to 1 as the absolute temperature T goes to 0. The equation for / for noninteracting bosons in a cubic box is/ = 1 - for T [McQuarrie (2000), Section 10-4]. 

The phenomenon of a significant fraction of bosons falling into the ground state is called Bose-Einstein condensation. Bose-Einstein condensation is important in determining the properties of superfluid liquid He (whose atoms are bosons), but the interatomic interactions in the liquid make theoretical analysis difficult. 

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The so-ealled Slater-Condon rules express the matrix elements of any one-eleetron (F) plus two-eleetron (G) additive operator between pairs of antisymmetrized spin-orbital produets that have been arranged (by permuting spin-orbital ordering) to be in so-ealled maximal eoineidenee. Onee in this order, the matrix elements between two sueh Slater determinants (labelled >and are summarized as follows ... 

In practice, each CSF is a Slater determinant of molecular orbitals, which are divided into three types inactive (doubly occupied), virtual (unoccupied), and active (variable occupancy). The active orbitals are used to build up the various CSFs, and so introduce flexibility into the wave function by including configurations that can describe different situations. Approximate electronic-state wave functions are then provided by the eigenfunctions of the electronic Flamiltonian in the CSF basis. This contrasts to standard FIF theory in which only a single determinant is used, without active orbitals. The use of CSFs, gives the MCSCF wave function a structure that can be interpreted using chemical pictures of electronic configurations [229]. An interpretation in terms of valence bond sti uctures has also been developed, which is very useful for description of a chemical process (see the appendix in [230] and references cited therein). 

Expanding the Slater determinants and integrating out the spin part and collecting terms that are the same under exchange of electeon indices, we have... 

To this pom t, th e basic approxmi alien is th at th e total wave I lnic-tion IS a single Slater determinant and the resultant expression of the molecular orbitals is a linear combination of atomic orbital basis functions (MO-LCAO). In other words, an ah miiio calculation can be initiated once a basis for the LCAO is chosen. Mathematically, any set of functions can be a basis for an ah mitio calculation. However, there are two main things to be considered m the choice of the basis. First one desires to use the most efficient and accurate functions possible, so that the expansion (equation (49) on page 222). will require the few esl possible term s for an accurate representation of a molecular orbital. The second one is the speed of tW O-electron integral calculation. 

V i hen the Slater determinant is expanded, a total of N1 terms results. This is because there are N different permutations of N eleefrons. For example, for a three-electron system with spin orbitals X2 and xs the determinant is... 

For any sizeable system the Slater determinant can be tedious to write out, let alone the equivalent full orbital expansion, and so it is common to use a shorthand notation. Various notation systems have been devised. In one system the terms along the diagonal of the matrix are written as a single-row determinant. For the 3x3 determinant we therefore have ... 

VVc can now see why the normalisation factor of the Slater determinantal wavefunction is I v/N . If each determinant contains N terms then the product of two Slater determinants, ldeU rminant][determinant], contains (N ) terms. However, if the spin orbitals form an oi lhonormal set then oidy products of identical terms from the determinant will be nonzero when integrated over all space. We Ccm illustrate this with the three-electron example, k ljiiiidering just the first two terms in the expansion we obtain the following ... 

Ihe one-electron orbitals are commonly called basis functions and often correspond to he atomic orbitals. We will label the basis functions with the Greek letters n, v, A and a. n the case of Equation (2.144) there are K basis functions and we should therefore xpect to derive a total of K molecular orbitals (although not all of these will necessarily 3e occupied by electrons). The smallest number of basis functions for a molecular system vill be that which can just accommodate all the electrons in the molecule. More sophisti- ated calculations use more basis functions than a minimal set. At the Hartree-Fock limit he energy of the system can be reduced no further by the addition of any more basis unctions however, it may be possible to lower the energy below the Hartree-Fock limit ay using a functional form of the wavefunction that is more extensive than the single Slater determinant. 

In accordance with the variation theorem we require the set of coefficients that gives the lowest-energy wavefunction, and some scheme for changing the coefficients to derive that wavefunction. For a given basis set and a given functional form of the wavefunction (i.e. a Slater determinant) the best set of coefficients is that for which the energy is a minimum, at which point... 

In order to calculate higher-order wavefunctions we need to establish the form of the perturbation, f. This is the difference between the real Hamiltonian and the zeroth-order Hamiltonian, Remember that the Slater determinant description, based on an orbital picture of the molecule, is only an approximation. The true Hamiltonian is equal to the sum of the nuclear attraction terms and electron repulsion terms ... 

Equation (3.74) is the exact exchange energy (obtained from the Slater determinant the Kohn-Sham orbitals), is the exchange energy under the local spin densit) ... 

The Slater determinant changes sign on exchange of any two rows (elections), so it satisfies the principle of antisymmetiical fermion exchange. 

The anti symmetrized orbital produet A (l)i(l)2Cl)3 is represented by the short hand (1>1(1>2(1>3 I and is referred to as a Slater determinant. The origin of this notation ean be made elear by noting that (1/VN ) times the determinant of a matrix whose rows are labeled by the index i of the spin-orbital (jii and whose eolumns are labeled by the index j of the eleetron at rj is equal to the above funetion A (l)i(l)2Cl)3 = (1/V3 ) det(( )i (rj)). The general strueture of sueh Slater determinants is illustrated below ... 

We substitute these expressions into the Slater determinants that form the singlet and triplet states and eolleet terms and throw out terms for whieh the determinants vanish. 

To express, in terms of Slater determinants, the wavefunctions corresponding to each of the states in each of the levels, one proceeds as follows ... 

One more quantum number, that relating to the inversion (i) symmetry operator ean be used in atomie eases beeause the total potential energy V is unehanged when all of the eleetrons have their position veetors subjeeted to inversion (i r = -r). This quantum number is straightforward to determine. Beeause eaeh L, S, Ml, Ms, H state diseussed above eonsist of a few (or, in the ease of eonfiguration interaetion several) symmetry adapted eombinations of Slater determinant funetions, the effeet of the inversion operator on sueh a wavefunetion P ean be determined by ... 

In faet, the Slater determinants themselves also are orthonormal funetions of N eleetrons whenever orthonormal spin-orbitals are used to form the determinants. 

The full N terms that arise in the N-eleetron Slater determinants do not have to be treated explieitly, nor do the N (N + l)/2 Hamiltonian matrix elements among the N terms of one Slater determinant and the N terms of the same or another Slater determinant. 

The resultant family of six eleetronie states ean be deseribed in terms of the six eonfiguration state funetions (CSFs) that arise when one oeeupies the pair of bonding a and antibonding a moleeular orbitals with two eleetrons. The CSFs are eombinations of Slater determinants formed to generate proper spin- and spatial symmetry- funetions. 

The spin- and spatial- symmetry adapted N-eleetron funetions referred to as CSFs ean be formed from one or more Slater determinants. For example, to deseribe the singlet CSF eorresponding to the elosed-shell orbital oeeupaney, a single Slater determinant... 

Also, the Ms = 1 eomponent of the triplet state having aa orbital oeeupaney ean be written as a single Slater determinant ... 

However, to deseribe the singlet CSF and Ms = 0 triplet CSF belonging to the aa oeeupaney, two Slater determinants are needed ... 

To simplify the analysis of the above CSFs, the familiar homonuelear ease in whieh z = 1.0 will be examined first. The proeess of substituting the above expressions for a and a into the Slater determinants that define the singlet and triplet CSFs ean be illustrated as follows ... 

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