Electronic Hamiltonian effective

These considerations are, of course, really bound up with the Hiickel theory s failure to deal with the electron-interaction terms which arise— l/rfJ for each pair of electrons, i and j. In the Hiickel scheme, an attempt is made to average such terms in order to obtain the effective, one-electron Hamiltonian, effective ( 2.2). It is during this averaging process that the singlet-triplet distinctions which we have been describing in this subsection are lost. 

Note that the one-electron Hamiltonian effective matrix components differ from those of Eq. (43) in what they truly represent. In this form, it represents the kinetic energy plus the interaction of a single electron with the core electrons around all nuclei present. The other integrals appearing in Eq. (52) are generally called two-electron-multi-centers integrals and are written as ... 

Note that now the one-electron Hamiltonian effective matrix components differ from those of Eq. (4.272) in what they truly represent, this time... 

The adiabatic picture is the standard one in quantum chemistry for the reason that, not only is it mathematically well defined, but it is also that used in ab initio calculations, which solve the electronic Hamiltonian at a particular nuclear geometry. To see the effects of vibronic coupling on the potential energy surfaces one must move to what is called a diabatic representation [1,65,180, 181]. 

In this work, relativistic effects are included in the no-pah or large component only approximation [13]. The total electronic Hamiltonian is H (r R) = H (r R) + H (r R), where H (r R) is the nom-elativistic Coulomb Hamiltonian and R) is a spin-orbit Hamiltonian. The relativistic (nomelativistic) eigenstates, are eigenfunctions of R)(H (r R)). Lower (upper)... 

While the equations of the Hartree-Fock approach can he rigorously derived, we present them post hoc and give a physical description of the approximations leading to them. The Hartree-Fock method introduces an effective one-electron Hamiltonian. as in equation (47) on page 194 ... 

I h e preceding discussion mean s that tli e Matrix etjuatiori s already described are correct, except that the Fuck matrix, F. replaces the effective one-electron Hamiltonian matrix, and th at K depends on th e solution C ... 

Ihe Fock operator is an effective one-electron Hamiltonian for the electron in the poly-tiectronic system. However, written in this form of Equation (2.130), the Hartree-Fock... 

The principal semi-empirical schemes usually involve one of two approaches. The first uses an effective one-electron Hamiltonian, where the Hamiltonian matrix elements are given empirical or semi-empirical values to try to correlate the results of calculations with experiment, but no specified and clear mathematical derivation of the explicit form of this one-electron Hamiltonian is available beyond that given above. The extended Hiickel calculations are of this type. 

Aj[ the beginning of this chapter, I introduced the notion that the 16 electrons iU ethene could be divided conceptually into two sets, the 14 a and the 2 n electrons. Let me refer to the space and spin variables as xi, Xj, > xi6, and for the minute I will formally label electrons 1 and 2 as the 7r-electrons, with 3 through 16 the cr-electrons. Methods such as Huckel rr-electron theory aim to treat the TT-electrons in an effective field due to the nuclei and the remaining a electrons. To see how this might be done, let s look at the electronic Hamiltonian end see if it can be sensibly partitioned into a rr-electron part (electrons 1 and 2) and a cr part (electrons 3 through 16). We have... 

The Hamiltonian is an effective electronic Hamiltonian that operates on h ... 

In the weak-coupling limit unit cell a (, 0 7a for fra/u-polyacetylene) and the Peierls gap has a strong effect only on the electron states close to the Fermi energy eF-0, i.e., stales with wave vectors close to . The interaction of these electronic states with the lattice may then be described by a continuum, model [5, 6]. In this description, the electron Hamiltonian (Eq. (3.3)) takes the form ... 

If we except the Density Functional Theory and Coupled Clusters treatments (see, for example, reference [1] and references therein), the Configuration Interaction (Cl) and the Many-Body-Perturbation-Theory (MBPT) [2] approaches are the most widely-used methods to deal with the correlation problem in computational chemistry. The MBPT approach based on an HF-SCF (Hartree-Fock Self-Consistent Field) single reference taking RHF (Restricted Hartree-Fock) [3] or UHF (Unrestricted Hartree-Fock ) orbitals [4-6] has been particularly developed, at various order of perturbation n, leading to the widespread MPw or UMPw treatments when a Moller-Plesset (MP) partition of the electronic Hamiltonian is considered [7]. The implementation of such methods in various codes and the large distribution of some of them as black boxes make the MPn theories a common way for the non-specialist to tentatively include, with more or less relevancy, correlation effects in the calculations. 

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. 

However, there also exists a third possibility. By using a famous relation due to Dirac, the relativistic effects can be (in a nonunique way) divided into spin-independent and spin-dependent terms. The former are collectively called scalar relativistic effects and the latter are subsumed under the name spin-orbit coupling (SOC). The scalar relativistic effects can be straightforwardly included in the one-electron Hamiltonian operator h. Unless the investigated elements are very heavy, this recovers the major part of the distortion of the orbitals due to relativity. The SOC terms may be treated in a second step by perturbation theory. This is the preferred way of approaching molecular properties and only breaks down in the presence of very heavy elements or near degeneracy of the investigated electronic state. 

H being the effective 1-electron Hamiltonian operator and Ek the energy of the kth state. Inserting (1.1) in (1.2) leads to... 

If one has determined the operator 7 by a method which does not simultaneously determine the CMO s, then Eq. (26) can be looked upon as a one-electron Schroedinger equation to be solved for the CMO s. In this sense, the Fock operator can be thought of as an effective one-electron hamiltonian. Thus, a one-electron variational problem can be set up namely, we require... 

The indices are all defined in terms of the Hiickel molecular orbital method. This has been described on many occasions, and need not be discussed in detail here, but a brief statement of the basic equations is a necessary foundation for later sections. The method utilizes a one-electron model in which each tt electron moves in a effective field due partly to the a-bonded framework and partly to its averaged interaction with the other tt electrons. This corresponds conceptually to the Hartree-Fock approach (Section IX) but at this level no attempt is made to define more precisely the one-electron Hamiltonian h which contains the effective field. Instead, each 7r-type molecular orbital (MO) is approxi-... 

Tully has discussed how the classical-path method, used originally for gas-phase collisions, can be applied to the study of atom-surface collisions. It is assumed that the motion of the atomic nucleus is associated with an effective potential energy surface and can be treated classically, thus leading to a classical trajectory R(t). The total Hamiltonian for the system can then be reduced to one for electronic motion only, associated with an electronic Hamiltonian Jf(R) = Jf t) which, as indicated, depends parametrically on the nuclear position and through that on time. Therefore, the problem becomes one of solving a time-dependent Schrodinger equation ... 

In our non-BO calculations performed so far, we have considered atomic systems with only -electrons and molecular systems with only a-electrons. The atomic non-BO calculations are much less complicated than the molecular calculations. After separation of the center-of-mass motion from the Hamiltonian and placing the atom nucleus in the center of the coordinate system, the internal Hamiltonian describes the motion of light pseudoelectrons in the central field on a positive charge (the charge of the nucleus) located in the origin of the internal coordinate system. Thus the basis functions in this case have to be able to accurately describe only the electronic correlation effect and the spherically symmetric distribution of the electrons around the central positive charge. 

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