Spin Orbitals and Slater Determinants

In quantum mechanics, to completely describe an electron, the spin needs to be considered in addition to its motion in space. Electrons have a spin j =,  

From a set of K spatial orbitals, a set of 2K spin orbitals can be generated. In addition, if the spatial orbitals are orthonormal, so are the spin orbitals. In the rest of this chapter, the spin orbitals will be called molecular orbitals (MOs). 

The wavefunction of a iV-electron system must be antisymmetric with respect to the exchange of two electrons 

A wavefunction satisfying this requirement can be conveniently written in the form of a Slater determinant [12] 

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It is worth recalling that the classification of electron correlation excitations in terms of spin orbitals and Slater determinants does not correspond term by term to the classification that emerges when symmetry-adapted orbitals and SACs are used. The arguments... 

The ground-state Hartree-Fock wave function o is the Slater determinant mi 2 m of spin-orbitals. This Slater determinant is an antisymmetrized product of the spin-orbitals [for example, see Eq. (10.36)] and, when expanded, is the sum of n terms, where each term involves a different permutation of the electrqns among the spin-orbitals. Each term in the expansion of s an eigenfunction of the MP H for example, for a four-electron system, application of H to a typical term in the I>o expansion gives... 

It was pointed out in Chapter 5 that, when the independent electron approximation [Eqs. (8-8)-(8-ll)] is taken, all states belonging to the same configuration become degenerate. In other words, considerations of space-spin symmetry do not affect the energy in that approximation. Therefore, the HMO method can make no explicit use of spin orbitals or Slater determinants, and so il/jt is normally taken to be a single product function as in Eq. (8-8). The Pauli principle is provided for by assigning no more than two electrons to a single MO. 

As a first step in applying these rules, one must examine > and > and determine by how many (if any) spin-orbitals > and > differ. In so doing, one may have to reorder the spin-orbitals in one of the determinants to aehieve maximal eoineidenee with those in the other determinant it is essential to keep traek of the number of permutations ( Np) that one makes in aehieving maximal eoineidenee. The results of the Slater-Condon rules given below are then multiplied by (-l) p to obtain the matrix elements between the original > and >. The final result does not depend on whether one ehooses to permute ... 

As a result, the exaet CC equations are quartic equations for the ti , ti gte. amplitudes. Although it is a rather formidable task to evaluate all of the eommutator matrix elements appearing in the above CC equations, it ean be and has been done (the referenees given above to Purvis and Bartlett are espeeially relevant in this eontext). The result is to express eaeh sueh matrix element, via the Slater-Condon rules, in terms of one- and two-eleetron integrals over the spin-orbitals used in determining , ineluding those in itself and the Virtual orbitals not in . 

In open-shell electronic states, the orbitals are not all doubly occupied, and the preceding procedure is not applicable. However, if the wave function can be written as a single Slater determinant, one can use a modified procedure to obtain energy-localized MOs here also. The procedure is to deal with the a spin-orbitals and the jS spin-orbitals separately, using two different unitary transformation matrices Ba and B in (2.85). 

Since the number of possible Slater determinants is (jj), this again gives an exponential dependence on N. For example, the simplest chemically reasonable orbital 21 basis set for benzene has 72 spin orbitals and (4g) 10. Clearly this expansion method is feasible only if very few of the Slater determinants actually contribute to each of the first few wavefunctions. Hence a method is required for constructing the orbitals < >i so that it is known in advance that relatively few of the 4 will be important. 

Let 0i(r,a), i = 1, m denote a basis of m orthonormal spin orbitals where r are the spatial coordinates and o is the spin coordinate. A Slater determinant is an antisymmetric linear combination of one or more of these spin orbitals. The occupation of a given Slater determinant can be written as an occupation number vector, ln>, where nj is one if spin orbital j is occupied in the Slater determinant and nj is zero if spin orbital (j> is unoccupied. 

One particular advantage of Slater determinants constructed from orthonormsd spin-orbitals is that matrix elements between determinants over operators such as H sure very simple. Only three distinct cases arise, as is well known and treated elsewhere. It is perhaps not surprising that the simplest matrix element formulas should be obtained from the treatment that exploits symmetry the least, as only the fermion antisymmetry has been accounted for in the determinants. As more symmetry is introduced, the formulas become more complicated. On the other hand, the symmetry reduces the dimension of the problem more and more, because selection rules eliminate more terms. We consider here the spin adaptation of Slater determinants. 

The 16 spin orbitals in this determinant are the Kohn-Sham spin orbitals of the reference system each is the product of a Kohn-Sham spatial orbital y/i and a spin function a or jS. Equation 7.18 can be written in terms of the spatial KS orbitals by invoking a set of rules (the Slater-Condon rules [34]) for simplifying integrals involving Slater determinants ... 

From this definition it is evident that application of Yj to the Fermi vacuum is equivalent to annihilation of a particle (or creation of a hole) in 14>0 >. The effect of YA on the Fermi vacuum state is the creation of a particle (or annihilation of a hole) in I 0>. The effect of YA" on the Fermi vacuum is the creation of a particle in the virtual spin-orbitals and finally, the effect of YA" is the annihilation of a particle in virtual spin-orbitals. Thus e.g., a singly excited Slater determinant I ) can be described as... 

To illustrate the modifications of UHF formalism, it is convenient to consider pure spin symmetry for a single Slater determinant with Nc doubly occupied spatial orbitals Xi and N0 singly occupied orbitals y". The corresponding UHF state has Na mj = occupied spin orbitals and Np rns = — J, occupied spin orbitals f. The number of open-shell and closed-shell orbitals are, respectively Na = Na — Np > 0 and Nc = Np. Occupation numbers for the spatial orbitals are nc = 2, n ° = 1. If all orbital functions are normalized, a canonical form of the RHF reference state is defined by orthogonalizing the closed- and open-shell sets separately. 

A straightforward application of Slater s rules to the determinantal matrix elements, Eq. (5-14), shows that the renormalized orbital matrix elements become configuration dependent in the following way [c.f. the discussion following Eq. (5-8)]. Consider for simplicity the case in which the determinants u and u differ in one spin-orbital, i.e. u = Uui, u = Uu[, where Ui, u[ are spin-orbitals, and f/ is a determinantal function for (iVtT-l) electrons then the renormalized matrix elements off-diagonal in one spin-orbital can be taken to have the form (Freed ),... 

Here Dv is the determinant obtained from D when the spin orbital t / is replaced by the two spin orbitals and ip,. Using Slater s rules we can express these quantities in terms of the spin orbitals ... 

The molecular spin-orbitals (r) are chosen from a set of orthogonal molecular spin-orbitals i/c(F). Every Slater determinant is built up using different members of the complete set of molecular orbitals. The Slater determinants are referred to as configurations, and a wavefunction such as that in (15) is called a configuration interaction wavefunction. 

It has been shown [14] that a complete set of such T 1) operators consists of the union of sets of operators p that add an electron to a spin-orbital 4>p, operators p q a that add an electron to (f>p and excite another electron from to (f>g, operators p q r ab that add an electron to (f>p, excite an electron from to (f> and excite another electron from (f>j, to (t>g as well as higher-level electron addition and excitation operators up to the highest-level operators that add an electron and induce N excitations. In labeling these operators, the indices a, b, c, d, etc. are used to denote spin-orbitals occupied in a so-called reference Slater determinant within l0,A) and p,q,r,s, etc. are used to denote unoccupied (i.e. virtual) spin-orbitals. The reference determinant, which is what defines the concept of occupied and unoccupied spin-orbitals, is usually chosen to be the determinant lO) within the neutral molecule wave function... 

The number of Slater determinants generated from 2m available spin-orbitals and n electrons is... 

As before, xi (1) is used to indicate a fxmction that depends on the space and spin coordinates of the electron labelled 1. The factor l/ /M ensures that the wavefunction is normalised we shall see later why the normalisation factor has this particular value. This functional form of the wavefunction is called a Slater determinant and is the simplest form of an orbital wave-fxmction that satisfies the antisymmetry principle. The Slater determinant is a particularly convenient and concise way to represent the wavefunction due to the special properties of determinants. Exchanging any two rows of a determinant, a process which corresponds to exchanging two electrons, changes the sign of the determinant and therefore directly leads to the antisymmetry property. If any two rows of a determinant are identical, which would correspond to two electrons being assigned to the same spin orbital, then the determinant vanishes. This can be considered a manifestation of the Pauli principle, which states that no two electrons can have the same set of quantum numbers. The Pauli principle also leads to the notion that each spatial orbital can accommodate two electrons of opposite spins. 

The more accurate Hartree-Fock method approximates the wave function as an antisymmetrized product (Slater determinant or determinants) of one-electron spin-orbitals and finds the best possible forms for the spatial orbitals in the spin-orbitals. Hartree-Fock calculations are usually done by expanding each orbital as a linear combination of basis functions and iteratively solving the Hartree-Fock equations (11.12). The Slater-type orbitals (11.14) are often used as the basis functions in atomic calculations. The difference between the exact nonrelativistic energy and the Hartree-Fock energy is the correlation energy of the atom (or molecule). 

Fast computers led the interest of many researchers to general many-electron systems like Cl expansions based on an orbital description and Slater determinants. The main advantage of these methods is the reduction of n-electron Hamiltonian matrix elements to one- and two-electron integrals, as stated in the Slater-Condon rules, but also showing a slow convergence. There are two sources of the slow convergence of the Cl expansion. (1) The combinatorial problem . For an n-electron system and a basis of m spin-free one-electron functions the number of... 

It is possible to choose spin orbitals where the coefficients of the singly substituted determinant are equal to zero, even if doubly and higher substitutions are introduced. In a Cl type expansion using these orbitals, the Slater determinant with the latter spin orbitals has the maximum possible overlap with the true wave function. The new spin orbitals are therefore called best overlap orbitals. In a Slater determinant O, these orbitals minimize... 

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