Hamiltonian operators density functional theory

It is also of interest to study the "inverse" problem. If something is known about the symmetry properties of the density or the (first order) density matrix, what can be said about the symmetry properties of the corresponding wave functions In a one electron problem the effective Hamiltonian is constructed either from the density [in density functional theories] or from the full first order density matrix [in Hartree-Fock type theories]. If the density or density matrix is invariant under all the operations of a space CToup, the effective one electron Hamiltonian commutes with all those elements. Consequently the eigenfunctions of the Hamiltonian transform under these operations according to the irreducible representations of the space group. We have a scheme which is selfconsistent with respect to symmetty. 

Addition of these two inequalities gives E o + Eq> E o + Eq, showing that the assumption was wrong. In other words, for the ground state there is a one-to-one correspondence between the electron density and the nuclear potential, and thereby also with the Hamiltonian operator and the energy. In the language of density functional theory, the energy is a unique functional of the electron density, E[p]. 

Substituting the Hamiltonian operator H into Eq. 6.8, we obtain a possible definition for the atomic energies. In the case of Density Functional Theory, the operators can be formulated as follows. 

In Section 3.2.1, we discussed the two main sets of approximations available to us when considering the interaction terms present in the Hamiltonian operator. The first is ab initio theory, which has as its basis Hartree-Fock theory the second is density functional theory, which recasts the basic equations in terms of the electron density rather than the wavefunction directly. 

These are important points for any quantitative work, electron-electron interactions must be taken into account, and the theories underpinning computation of MOs do this at various levels of accuracy. Approaches such as Hartree-Fock or density functional theory adapt the Hamiltonian operator to include electron-electron terms in an averaged way so electrons see the Coulomb field of each other averaged over the calculated density associated with each MO (see the Further Reading section in this chapter). 

This book is concerned with the quantum chemical methods for the calculations of electromagnetic properties of molecules. However, in detail only so-called ab initio quantum chemical methods will be discussed in Part III. As ab initio methods one normally describes those quantmn chemical methods that start from the beginning, i.e. methods that require the evaluation of all the terms in the Schrodinger or Dirac equation and do not include other experimentally determined quantities than the nuclear charges, nuclear masses, nuclear dipole and quadrupole moments and maybe positions of the nuclei. This is in contrast to the so-called semi-empirical methods where many of the integrals over the operators in the Hamiltonian are replaced by experimentally or otherwise determined constants. However, in the case of density functional theory (DFT) methods the classification is somewhat debatable. 

Thus, let us assume that two different external potentials can each be consistent with the same nondegenerate ground-state density po- We will call these two potentials Va and wj, and the different Hamiltonian operators in which they appear and Ht,. With each Hamiltonian will be associated a ground-state wave function Pq and its associated eigenvalue Eq. The variational theorem of molecular orbital theory dictates that the expectation value of the Hamiltonian a over the wave function b must be higher than the ground-state energy of a, i.e.. 

One of our main motivations for pursuing the development of a density functional response theory for open-shell systems has been to calculate spln-Hamiltonian parameters which are fundamental to experimental magnetic resonance spectroscopy. It is only within the context of a state with well-defined spin we can speak of effective spin Hamiltonians. The relationship between microscopic and effective Hamiltonians rely on the Wigner-Eckart theorem for tensor operators of a specific rank and states which transform according to their irreducible representations [45]. 

A deeper argument is that local density functional derivatives appear to be implied by functional analysis [2,21,22]. The KS density function has an orbital structure, p = Y.i niPi = X fa- For a density functional Fs, strictly defined only for normalized ground states, functional analysis implies the existence of functional derivatives of the form SFj/ Sp, = e, — v(r), where the constants e, are undetermined. On extending the strict ground-state theory to an OFT in which OEL equations can be derived, these constants are determined and are just the eigenvalues of the one-electron effective Hamiltonian. Since they differ for each different orbital energy level, the implied functional derivative depends on a direction in the function-space of densities. Such a Gateaux derivative [1,2] is equivalent in the DFT context to a linear operator that acts on orbital functions [23]. 

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