Excited states matrix Hartree-Fock

The ground and excited state matrix Hartree-Fock energies for the He, Li and Be atoms are presented in Tables 1, 2 and 3, respectively. All energies are given in atomic units, (Hartree). In each of these tables, we label the columns according to the three schemes, (a), (b) and (c), described above for generating sequences of even-tempered basis sets. We consider each system in turn. 

There is far less reported experience for ab initio studies of electronically excited states than for ground states. Matrix Hartree-Fock calculations for excited states cannot be considered routine. Often the same basis set is used for both the ground and excited state even though as long ago as 1958 Shull and Lowdin [17] demonstrated... 

MATRIX HARTREE-FOCK ENERGIES FOR EXCITED STATES... 

A more general way to treat systems having an odd number of electrons, and certain electronically excited states of other systems, is to let the individual HF orbitals become singly occupied, as in Figure 6.3. In standard HF theory, we constrain the wavefunction so that every HF orbital is doubly occupied. The idea of unrestricted Hartree-Fock (UHF) theory is to allow the a and yS electrons to have different spatial wavefunctions. In the LCAO variant of UHF theory, we seek LCAO coefficients for the a spin and yS spin orbitals separately. These are determined from coupled matrix eigenvalue problems that are very similar to the closed-shell case. 

In a recent publication we have investigated this first order approximation to the particle-hole self energy for the choice y>) = o ) for the reference state tp) and starting from a Hartree-Fock zeroth order [21]. This particular approximation to the particle-hole self energy is referred to as First Order Static Excitation Potential (FOSEP). In terms of the matrix elements of the Hamiltonian the FOSEP approximation of the primary block H reads... 

In a sense, CIS is a certain counterpart of the (1-electron) Hartree-Fock method for excited state. Indeed, the Hartree-Fock (HF) variational parameters can be packed into an 1-electron matrix, C, of the conventional occupied MO expansion coefficients. In turn, the CIS configurational coefficients comprise the matrix which is just the first variational derivative of the C-matrix. More exactly, given the ground state tV-electron Slater determinant ) = (1... A)). This ) is the antisymmetrized product of the spin-orbitals, the latter being the standard spinless spatial MOs, (p°), equipped with spin variables. As usually, MOs °) are... 

The CEO computation of electronic structure starts with molecular geometry, optimized using standard quantum chemical methods, or obtained from experimental X-ray diffraction or NMR data. For excited-state calculations, we usually use the INDO/S semiempirical Hamiltonian model (Section IIA) generated by the ZINDO code " however, other model Hamiltonians may be employed as well. The next step is to calculate the Hartree— Fock (HE) ground state density matrix. This density matrix and the Hamiltonian are the Input Into the CEO calculation. Solving the TDHE equation of motion (Appendix A) Involves the diagonalization of the Liouville operator (Section IIB) which is efficiently performed using Kiylov-space techniques e.g., IDSMA (Appendix C), Lanczos (Appendix D), or... 

Using the ground-state density matrix as an input, the CEO procedure - computes vertical transition energies and the relevant transition density matrices (denoted electronic normal modes ( v)mn = g c j Cn v ), which connect the optical response with the underlying electronic motions. Each electronic transition between the ground state and an electronically excited state v) is described by a mode which is represented hy K x K matrix. These modes are computed directly as eigenmodes of the linearized time-dependent Hartree—Fock equations of motion for the density matrix (eq A4) of the molecule driven by the optical field. 

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