Singlet state, Hamiltonian matrix

In my discussion of pyridine, I took a combination of these determinants that had the correct singlet spin symmetry (that is, the combination that represented a singlet state). I could equally well have concentrated on the triplet states. In modem Cl calculations, we simply use all the raw Slater determinants. Such single determinants by themselves are not necessarily spin eigenfunctions, but provided we include them all we will get correct spin eigenfunctions on diago-nalization of the Hamiltonian matrix. 

Homosymmetric Biradicaloids The Hamiltonian matrix for the singlet state is... 

So far, the preceding considerations have been very useful for characterization of states and for ordering of singlet-state energies relative to (T) with fixed values of K and K. Nevertheless, we could not receive any qualitative ideas about the shapes of energy surfaces as the biradicaloids develop from the normal molecules. For this purpose the introduction of overlap, which we have neglected so far, is necessary. The Hamiltonian matrix for nonorthogonal orbitals A and B for nonsymmetric biradicaloid is... 

The spin singlet states of HDVV Hamiltonian are those forming the problem of PJT effect related to the bond alternation. Using the symmetry labels of D4h one may write the following model Hamiltonian whose non-diagonal matrix element... 

The residual interaction between these same-n Rydberg levels, expressed in the case (c) basis, but intermediate between cases (c) and (a), can now be viewed as a consequence of the electronic part of the Hamiltonian, since the case (c) basis functions are exact eigenfunctions only when the energy difference between isoconfigurational triplet and singlet states is zero. The off-diagonal matrix element in the case (c) basis is... 

Simultaneous presence of covalent and polarizing perturbations (tab AB ( ) leads to non-symmetric biradicaloids. Their general features again follow from a consideration of the Hamiltonian matrix in Table 5. For instance, they permit a ready rationalization of the preference of donor-acceptor pair molecules such as NH2-BH2 for planarity in the lowest singlet state Sq and orthogonality in the first excited singlet state Si [1,6]. 

A general wave function of a singlet state S of an electron pair is obtained by diagonalization of the matrix of the Hamiltonian H given in Table 5 ... 

Thus, if we let F be the field from the other molecule, we find that the matrix element (ai jb) of Eqnation 14.10 can be represented as in Eqnation 14.38. This is the well-known expression for the interaction between two dipole moments. A quite remarkable thing is that this is also the Hamiltonian matrix element in the Schrodinger equation for the excited singlet states, if the dipole moment is to be replaced by the transition dipole moment. Next, we will find that the rate of transfer of the Si state to the site is proportional to the square of this matrix element. 

If the Kramers partners are in different rows of the same, doubly degenerate irrep, as in the group then the matrix elements are real and the gradient is zero. Likewise, the Hamiltonian matrix element between the two wave functions is zero, because the two configurations span two boson irreps and the -I- and - linear combinations have specific boson symmetries. This situation is somewhat similar to the nonrelativistic case of and I states, which are linear combinations of two determinants, or to the open-shell singlet case. 

The matrix element (H d ) is determined according to (5.4.3) and (5.4.4) in terms of 1- and 2-electron transition densities, and may be obtained in a precisely similar manner (McWeeny and Sutcliffe, 1963). Since, however, it is usual to employ a spinless Hamiltonian, further reductions will be necessary, and these are less straightforward. Two new factors must be recognized (i) even when the functions, 0 are spatially non-degenerate, they may appear in spin-degenerate sets (e.g. triplets, quintuplets,.. . ), and the excited functions for AB will then be obtained by vector coupling -, and (ii) when non-singlet states occur the further reduction will also lead to spin densities. 

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