Antisymmetric space factor

Both of these space factors correspond to Mp = 1. The symmetric space factor combines with the antisymmetric spin factor, leading to one state with S = 0. The three triplet spin factors are all symmetric, so the antisymmetric space factor can combine with any of these, leading to three states, with Ms equal to 1,0, and —1, corresponding to 5 = 1. The 2p0 orbital combines with the li orbital in exactly the same way as the 2pl orbital except that Mp = 0 for these wave functions. The 2p,— orbital combines with the li orbital in exactly the same way as the 2p orbital except that Mp = — 1. This gives a total of 12 states. 

The three values of Mi, correspond to L = 1 with no states leftover, so that only P terms occur. Each symmetric spin factor combines with each one of the three antisymmetric space factors to give the nine states of the P term, and the antisymmetric spin factor combines with each one of the symmetric space factors to give the three states of the P term. 

An antisymmetrized orbital wave function can also exhibit electron correlation if it has an antisymmetric space factor like that of Eq. (18.4-2a). Consider the wave function... 

The wavefunction (1), antisymmetric under exchange of space-spin variables of the two electrons, was conveniently factorized into a product of space and spin factors anti-parallel coupling of the spins (antisymmetric spin factor) then implied symmetry of the spatial factor and a consequent enhancement of electron density in the bond region. Such a factorization is unique to a 2-electron system for an 7V-electron system, a Pauli-compatible function must have the form... 

If the electrons occupy different space orbitals there are four possible states for each pair of space orbitals. For the (ls)(25) configuration we can write four antisymmetric wave functions that are products of a space factor and a spin factor ... 

To generalize the method of Heitler and London, we consider any configuration containing N singly occupied orbitals , 2, normally valence AOs, analogous to the Is orbitals in the H2 calculation, but for the moment are assumed orthogonal. Spin factors may then be allocated in 2 possible ways. We now set up antisymmetric space-spin functions as in Section 4.2 (p. 88), starting from an orbital function... 

The study of the two-electron systems was greatly simplified by the fact that the total wave function could be factorized into a space part and a spin part according to Eq. III. I. ForiV = 3, 4,. . , such a separation of space and spin is no longer possible, and an explicit treatment of the spin is actually needed in considering correlation effects. This question of the connection between space and spin in an antisymmetric spin function is a rather complicated problem, which has been brought to a simple solution first during the last few years. 

The factor (l/ /2) is inserted so that the total probability added over all configurations is unity. This function may be said to be symmetric in the space co-ordinates but antisymmetric in the spins. For an overall interchange it is antisymmetric. 

The l/v/4 factor ensures that the wavefunction is normalized, i.e. that IT)2 integrated over all space = 1 [11]). This Slater determinant ensures that there are no more than two electrons in each spatial orbital, since for each spatial orbital there are only two 1-electron spin functions, and it ensures that F is antisymmetric since switching two electrons amounts to exchanging two rows of the determinant, and this changes its sign (Section 4.3.3). Note that instead of assigning the electrons successively to row 1, row 2, etc., we could have placed them in column 1, column 2, etc. of... 

For one- or two-electron wavefunctions the space and spin parts can be factored. Assume that one of the electrons is in electronic state m, the other in electronic state n. Then one can write antisymmetrized wavefunctions of the type... 

Here, the space-spin function is a one-term product of a spatial part and a spin factor. The overall function is antisymmetric, the spin factor is antisymmetric with respect to exchange of spin coordinates thus the spatial part must be symmetric with respect to exchange of spatial coordinates. Having used this formalism to ensure that the total wavefunction is antisymmetric, we are finished with spin completely. This result is clearly independent of the specific form of the HL trial function and ... 

We have in fact to compare the energy of an APG function (Antisymmetrized Product of Geminals, without strong orthogonality) with the sum of individual pair energies (without factorization of the Hilbert space). The difference... 

These equations are expressed in the spin-orbital formalism and the products of orbitals are assumed to be antisymmetrized. The coefficients are the explicitly correlated analogues of the conventional amplitudes. The xy indices refer to the space of geminal replacements which is usually spanned by the occupied orbitals. The operator Q12 in Eq. (21) is the strong orthogonality projector and /12 is the correlation factor. In Eq. (18) the /12 correlation factor was chosen as linear ri2 term. It is not necessary to use it in such form. Recent advances in R12 theory have shown that Slater-type correlation factors, referred here as /12, are advantageous. Depending on the choice of the Ansatz of the wave function, the formula for the projector varies, but the detailed discussion of these issues is postponed until Subsection 4.2. The minimization of the Hylleraas functional... 

This matrix may be factorized by spin and space symmetries into a onedimensional triplet antisymmetric, a one-dimensional singlet antisymmetric and a two-dimensional singlet symmetric subspaces... 

In discussing the helium atom (Section 1.2) the antisymmetry requirement on the electronic wavefunction was easily satisfied for with only two electrons the function would be written as a product of space and spin factors, one of which had to be antisymmetric, the other symmetric, lliis is possible even for an exact eigenfunction of the Hamiltonian (1.2.1), as well as for an orbital product. The construction of an antisymmetric many-electron function is less easy. We have seen in Section 1.2 that for a general permutation (involving both space and spin variables) an antisymmetric function has the property... 

In the development of the Slater method (Section 3.1) it was noted that the Pauli principle in the form (1.2.27) could always be satisfied by constructing the electronic wavefunction from determinants (i.e. antisymmetrized products) of spin-orbitals. In an earlier section, however, it was shown that for a two-electron system the antisymmetry principle could also be satisfied by writing the wavefunction as a product of individually symmetric or antisymmetric factors—one for spatial variables and the other for spin variables. Since, in the usual first approximation the Hamiltonian does not contain spin variables, it is natural to enquire whether a corresponding exact N-electron wavefunction might be written as a space-spin product in which the spatial factor is an exact eigenfunction of the spinless Hamiltonian (1.2.1). To investigate this possibility, we need a few basic ideas from group theory (Appendix 3). 

Find a Schrodinger operator equivalent to the Fock-space creation operator Hence show, using the form of the destruction operator d, found on p. 462, that any creation/annihilation pair sudi as aja, will give a result that is independent of the number of electrons in the ket upon which it acts (in conformity with the properties established and used in various chapters). [Hint The operator will act on an A -electron function, adding an (N + l)th electron, and will be A -dependent. It must involve an antisymmetrizer for the N +1)-electron function and this may be factorized —see (14.1.4).]... 

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