Zero-order electronic Hamiltonian

In the zero-order approximation we ignore the electron-electron repulsion, represented by the term proportional to l/ri2. The zero-order electronic Hamiltonian operator is now a sum of two hydrogen-molecule-ion Hamiltonians ... 

In the Born-Oppenheimer approximation the zero-order electronic Hamiltonian operator for diatomic helium consists of four hydrogen-molecule-ion-like (HMIL) Hamiltonian operators ... 

The simplest stable heteronuclear molecule is lithium hydride, LiH. Like Hc2, LiH has two nuclei and four electrons. We say that these two molecules are isoelectronic (have the same number of electrons). However, LiH is a stable molecule in the ground state while He2 is not. Figure 20.13 shows the LiH system. We apply the Born-Oppenheimer approximation and place the nuclei on the z axis and place the origin of coordinates at the center of mass of the nuclei. The zero-order electronic Hamiltonian operator for the LiH molecule is (omitting the constant intemuclear repulsion termf ui)... 

In this case the zero-order electronic wave functions are, in principle, referred to a Hamiltonian that contains the potential from the ions at their actual positions, i.e., the electrons follow the ionic motion adiabatically. Since both these approximations are sometimes referred to as the Born Oppenheimer approximation, this has led to confusion in terminology for example, Mott (1977) refers to the Born-Oppenheimer approximation, but gives wave functions of the adiabatic type, whereas Englman (1972) differentiates between the two forms, but specifically calls the static form the Bom Oppenheimer method. [We note that, historically, the adiabatic form was first suggested by Seitz (1940)—see, for example, Markham (1956) or Haug and Sauermann (1958)]. In this chapter, we shall preferentially use the terminology static and adiabatic. [Note that the term crude adiabatic is also sometimes used for the static approximation, mainly in the chemical literature—see, for example, Englman (1972, 1979).]... 

We now employ perturbation theory to first calculate the electronic corrections to the zero-order energy due to the Hamiltonian in Eq. (6). Using the zero-order electronic functions in Eq. (4) leads to the following rotational Hamiltonian corrected to several orders in electronic states (the higher order terms are dropped)2 ... 

Since electrons are much faster than nuclei, owing to Wg Mj, ions can be considered as fixed and one can thus neglect the //ion-ion contribution (formally Mion-ion Hee, where Vion-ion is a Constant). This hrst approximation, as formulated by N. E. Born and J. R. Oppenheimer, reflects the instantaneous adaptation of electrons to atomic vibrations thus discarding any electron-phonon effects. Electron-phonon interactions can be a-posteriori included as a perturbation of the zero-order Hamiltonian Hq. This is particularly evident in the photoemission spectra of molecules in the gas phase, as already discussed in Section 1.1 for nJ, where the 7T state exhibits several lines separated by a constant quantized energy. 

Here, Q is the projector on the bound subspace and P projects onto the open, continuum channels. The intramolecular coupling is written as V+ U so that, as before, U is any additional coupling brough about by external perturbations. The equation H = Hq + V+U, where Ho is the zero-order Hamiltonian of the Rydberg electron and so includes only the central part of the potential due to the core plus the motion (vibration, rotation) of the core, uncoupled to the electron. The perturbations V + U can act within the bound subspace, as the operator Q(V+l/)Q is not necessarily diagonal and is the cause of any intramolecular dynamics even in the absence of coupling to the continuum. The intramolecular terms can also couple the bound and dissociative states. 

The zero-order, spin-free Hamiltonian HSF commutes with the symmetric group SnF of permutations on the N different spatial electronic indices,... 

If electrons are localized in different parts of a molecule, exchange terms between these electrons are small. In this localized state several different permutation states may be degenerate or near degenerate in zero order. Consequently, these states are highly mixed under the full Hamiltonian. 

An ab initio method is one that starts from a zero-order Hamiltonian that is exact except for relativistic and magnetic effects and that involves the evaluation of electronic energies and other relevant quantities for wave functions that are properly antisymmetrized in the coordinates of all the electrons. For a system containing n electrons and M nuclei, the zero-order Hamiltonian depends parametrically on the nuclear positions and is of the... 

In the approximation considered, the non-relativistic (zero-order) Hamiltonian Hq of the N-electron atom may be written as follows ... 

Fig. 2. Schematic potential curves showing avoided crossing in higher approximation. The full curves A and B are electronic eigenvalues (with r as parameter) for the complete Hamiltonian, neglecting coupling with nuclear motion. The zero order curves A and B0, including dashed parts through the crossing point, intersect because they are derived from an incomplete Hamiltonian. The arrow represents the possibility of spontaneous radiation from the A state to the...

First order terms in Eq. (1) due to vibronic coupling may in general give rise to changes of the electronic wavefunctions. It can be easily seen that eigenfunctions (q) of the zero-order Hamiltonian H(q,0) may be intermixed by first order perturbation yielding... 

Solution of the Schroedinger equation, H p = Ef, appropriate to this problem has only been accomplished by means of successive perturbation calculations. The zero-order approximation is a spherical approximation in which a given outer electron is assumed to move in the average potential of the other outer electrons as well as of the core electrons. Then the free-ion Hamiltonian becomes... 

In the above expressions the iV-electron Hartree-Fock model hamiltonian, o, was used as a zero-order operator. This leads to the perturbation series of the type first discussed through second-order by Moller and Plesset.55 84 However, it is clear that any operator X obeying the relation... 

In electronic structure calculations, it is not unlikely for a basis set to be dependent on the parameters. The most obvious case involves geometric parameters. The atomic orbital basis functions used to construct molecular orbitals are generally chosen to follow the atomic centers. This means that the functions are dependent on the molecular geometry, and so there will be nonzero derivatives of the usual one- and two-electron integrals. In the case of parameters such as an electric field strength, there is no functional dependence of the standard types of basis functions. The derivatives of all the basis functions with respect to this parameter are zero, and so all derivative integrals involving the zero-order Hamiltonian terms are zero as well. 

The Kronecker delta functions, 5 and 6,p, resulting from Eq. [21], cannot be simplified to 1 or 0 because the indices p and q may refer to either occupied or virtual orbitals. The important point here, however, is that the commutator has reduced the number of general-index second-quantized operators by one. Therefore, each nested commutator from the Hausdorff expansion of H and T serves to eliminate one of the electronic Hamiltonian s general-index annihilation or creation operators in favor of a simple delta function. Since f contains at most four such operators (in its two-electron component), all creation or annihilation operators arising from f will be eliminated beginning with the quadruply nested commutator in the Hausdorff expansion. All higher order terms will contain commutators of only the cluster operators, T, and are therefore zero. Hence, Eq. [52] truncates itself naturally after the first five terms on the right-hand side. ° This convenient property results entirely from the two-electron property of the Hamiltonian and from the fact that the cluster opera-... 

Strictly speaking, however, the spin angular momentum and its components are not constants of motion in nonlinear molecules, and the classification of states by multiplicity is therefore only approximate. Spin-orbit coupling is the most important of the terms in the Hamiltonian that cause a mixing of zero-order pure multiplicity states. The interaction between the spin angular momentum of an electron and the orbital angular momentum of the same electron causes the presence of a minor term in the Hamiltonian, which may be written as... 

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