Integration over spin

There are useful two- and many-electron analogues of the functions discussed above, but when the Hamiltonian contains only one- and two-body operators it is sufficient to consider the pair functions thus the analogue of p(x x ) is the pair density matrix 7t(xi,X2 x i,x ) while that of which follows on identifying and integrating over spin variables as in (4), is H(ri,r2 r i,r2)- When the electron-electron interaction is purely coulombic, only the diagonal element H(ri,r2) is required and the expression for the total interaction energy becomes... 

Note that in Eqs. (22), (23), and (25) r denotes spatial coordinates (integration over spin degrees of freedom has been omitted for the sake of brevity). 

Integration over spin functions simplifies this equation because the first two sums in Eq. (92) may be rewritten in terms of the number of a- and / -electrons /VQ,/and N, respectively,... 

In order for V to be nonzero, it is necessary that the integration over spin contains at least one nonzero component. Without elaborating on the details, this will only be the case if the change in spin at each redox site is AS = A as would be the case, for example, for Ru(NH3)63+/2+ self-exchange (equation 47a). 

In the previous sections, the occupation number vectors were specified in terms of the occupation of a set of spin orbitals, and the operators were defined by integrals over spin orbitals multiplied with spin orbital excitation operators. The spin orbitals depend on a continuous spatial coordinate, r, and a discrete spin coordinate ms. The spin coordinate takes two values, so the complete spin basis is spanned by two functions a(ms), a = a, p defined as... 

G. Gaigalas, Integration over spin-angular variables in atomic physics, Lithuanian J. Phys., 39, 79-105 (1999). 

In the simplest case, when the scattering system is a hydrogen atom, after integration over spin variables the exchange amplitude in the... 

The shift operators in the spin space act as unit operators (Eq.(20) corresponds to an integration over spin variables) and their action in the orbital space may be expressed in terms of the orbitals as [19]... 

In order to evaluate integrals over spin variables let us note that Eq. (5) implies that the symmetric group total spin operators S2 and S2. Consequently, the eigenfunctions of these operators form bases for representations of [Pg.614]

Some of the terms above may vanish after integrating over spin coordinates, and a pair of determinants differing by more than two spin orbitals have a matrix element of zero. A derivation of these rules can be found in the introductory text by Szabo and Ostlund.63 The rules for evaluating Hamiltonian matrix elements in a CSF basis are more complicated and are generally derived40,42,78 using second quantization, which we consider next. 

On the contrary, the dipole-dipole terms play no role at all for coupling constants, because they vanish after integration over spin coordinates, and exchange integrals only are significant in NMR. 

On removing the primes in equations (18) and (19) and integrating over spin, one obtains the spinless densities ... 

Integration over spin and space variables is denoted by... 

In the above equation and in what follows, integration over spin and space variables is denoted by... 

As usual, the electronic Hamiltonian does not include spin so, since the singlet spin function for two electrons is a separate factor, we may integrate over spin to give... 

The integrals over spin-orbitals (which include a sum over spins) are readily evaluated in terms of electron-repulsion integrals. The sums over a, b, i, and j in (15.87) provide for the inclusion of all the doubly substituted i/ff s in (15.86). 

To do an MP electron-correlation calculation, one first chooses a basis set and carries out an SCF calculation to obtain o, hf> and virtual orbitals. One then evaluates EP (and perhaps higher corrections) by evaluating the integrals over spin-orbitals in (15.87) in terms of integrals over the basis functions. One ought to use a complete set of basis functions to expand the spin-orbitals. The SCF calculation will then produce the exact Hartree-Fock energy and will yield an infinite number of virtual orbitals. The first two sums in (15.87) will then contain an infinite number of terms. Of course, one always uses a finite, incomplete basis set, which yields a finite number of virtual orbitals, and the sums in (15.87) contain only a finite number of terms. One thus has a basis-set truncation error in addition to the error due to truncation of the MP perturbation energy at E or E or whatever. 

Substitution of this result into Eq. 9.41 and integrating over spins yields... 

Before generalizing the above results and presenting general expressions for matrix elements involving N-electron determinants, it is appropriate to summarize the different notations we use in this book for one- and two-electron integrals. The notation for two-electron integrals over spin orbitals that we have introduced in Eq. (2.90), i.e.,... 

It is an unfortunate fact of life that there is another notation for two-electron integrals over spin orbitals in common use, particularly in the literature of Hartree-Fock theory. This notation, often referred to as the chemists notation, is... 

Table 2.2 Notations for one- and two-dectron integrals over spin (Z) and spatial ( ) orbitals...

For one-electron integrals over spin orbitals, the chemists and physicists notations are essentially the same. 

Table 2.2 summarizes all the notations for one- and two-electron integrals used in this book. When we consider the reduction of integrals over spin orbitals to integrals over spatial orbitals later in this chapter, we will introduce a new notation for spatial integrals, which we have included in the table for the sake of completeness and ease of future reference. 

The summation of antisymmetrized two-electron integrals is thus over all unique pairs of spin orbitals Xm Xn occupied in C>. This observation suggests a simple mnemonic device for writing down the energy of any single determinant in terms of one- and two-electron integrals over spin orbitals. Each occupied spin orbital Xi contributes a term to the energy and... 

If we now multiply this equation by a (coi) and integrate over spin we get... 

Exercise 3.33 Rather than use the simple technique of writing down f l) by inspection of the possible interactions, as we have done above, use expression (3.314) for / ( ) and explicitly integrate over spin and carry through the algebra, as was done in Subsection 3.4.1 for the restricted closed-shell case, to derive... 

Since 77(1,2) contains no spin operators at our level of approximation, the integral separates into an integral over the space coordinates of both electrons and an integral over the spin coordinates of both electrons. The integration over spins gives a factor of unity. There remains... 

We consider the integration over spin to have been carried out already, giving a factor of unity. Integrating over to obtain In, and regrouping terms gives... 

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