Hamiltonian diatomic electronic

The atomic SCF calculations described in Section 2.3 may be extended in principle to diatomic molecules with closed-shell electron configurations. The diatomic electronic Hamiltonian in the clamped-nuclei approximation (Eqs. 3.7 and 3.9) may be broken down into a sum of one-electron operators and electron repulsion terms... 

Just as for an atom, the hamiltonian H for a diatomic or polyatomic molecule is the sum of the kinetic energy T, or its quantum mechanical equivalent, and the potential energy V, as in Equation (1.20). In a molecule the kinetic energy T consists of contributions and from the motions of the electrons and nuclei, respectively. The potential energy comprises two terms, and F , due to coulombic repulsions between the electrons and between the nuclei, respectively, and a third term Fg , due to attractive forces between the electrons and nuclei, giving... 

In the general case R denotes a set of coordinates, and Ui(R) and Uf (R) are potential energy surfaces with a high dimension. However, the essential features can be understood from the simplest case, which is that of a diatomic molecule that loses one electron. Then Ui(R) is the potential energy curve for the ground state of the molecule, and Uf(R) that of the ion (see Fig. 19.2). If the ion is stable, which will be true for outer-sphere electron-transfer reactions, Uf(R) has a stable minimum, and its general shape will be similar to that of Ui(R). We can then apply the harmonic approximation to both states, so that the nuclear Hamiltonians Hi and Hf that correspond to Ui and Uf are sums of harmonic oscillator terms. To simplify the mathematics further, we make two additional assumptions ... 

A first insight into a different description of a chemical process can be obtained from an analysis of a (diatomic) dissociation process. Consider the standard treatment of a stable diatomic molecule. The word stable implies already the existence of a measurable characteristic size around which the electro-nuclear system fluctuates in its ground electronic state (i.e. a stationary molecular Hamiltonian with ground state). In standard quantum chemistry, this is the nuclear equilibrium distance. 

To illustrate this point, consider a composite system composed of two noninteracting subsystems, one with p electrons (subsystem A) and the other with q = N — p electrons (subsystem B). This would be the case, for example, in the limit that a diatomic molecule A—B is stretched to infinite bond distance. Because subsystems A and B are noninteracting, there must exist disjoint sets Ba and Bb of orthonormal spin orbitals, one set associated with each subsystem, such that the composite system s Hamiltonian matrix can be written as a direct sum. 

The electron diffraction study was complemented by an all-electron theoretical calculation of Lu, Wei, and Zunger (LWZ) (1992), using the local density approximation for the exchange and correlation terms in the Hamiltonian. They find agreement within x0.6% between the calculated and dynamic structure factor values for the lowest three reflections, (100), (110), and (111). But for (200), with sin 0/A = 0.3464 A-1, the discrepancy is as large as 1.7%. The discrepancy is attributed to insufficiently accurate knowledge of the temperature factors in this diatomic crystal, which affect the derivation of the X-ray structure factor from the electron diffraction measurement, as well as the calculation of the dynamic theoretical structure factors needed for the comparison with experiment. For the monoatomic Si crystal for which the B values are well known, the agreement is... 

The direct variational solution of the Schrddinger equation after separation of the center of mass motion is in general possible and can be performed very accurately for three- and four- body systems such as (Kolos, 1969) and H2 (Kolos and Wolniewicz, 1963 Bishop and Cheung, 1978). For larger systems it is unlikely to perform such calculations in the near future. Therefore the usual way in quantum chemistry is to introduce the adiabatic approximation. The nonrelativistic hamiltonian for a diatomic N-electron molecule in the center of mass system has the following form (in atomic units). 

There are many systems of different complexity ranging from diatomics to biomolecules (the sodium dimer, oxazine dye molecules, the reaction center of purple bacteria, the photoactive yellow protein, etc.) for which coherent oscillatory responses have been observed in the time and frequency gated (TFG) spontaneous emission (SE) spectra (see, e.g., [1] and references therein). In most cases, these oscillations are characterized by a single well-defined vibrational frequency, It is therefore logical to anticipate that a single optically active mode is responsible for these features, so that the description in terms of few-electronic-states-single-vibrational-mode system Hamiltonian may be appropriate. 

In Section 1.19 we classified the electronic wave functions of homonuclear diatomic molecules as g or u, according to whether they were even or odd with respect to inversion g and u refer to inversion of the electronic coordinates with respect to the molecule-fixed axes. This is to be distinguished from the inversion of electronic and nuclear coordinates with respect to space-fixed axes, which was discussed in this section. The electronic Hamiltonian for a diatomic molecule is... 

The complete, nonrelativistic Hamiltonian for a diatomic molecule is given by (1.272). If one inverts the Cartesian coordinates of all particles (nuclei and electrons), then H in (1.272) is unchanged, since all interparticle distances are unchanged. Thus the parity operator IT commutes with this Hamiltonian, and we can characterize the overall wave function of a diatomic molecule by its parity. (This statement applies to both homonuclear and heteronuclear diatomics.)... 

Atoms in Molecules.—In this approach, which was first proposed by Moffitt,105 a wavefunction for a particular electronic state of a molecule is constructed from products of atomic wavefunctions, these, moreover, being taken to be exact eigenfunctions of their respective atomic hamiltonians. We confine our attention to the case of diatomic molecules AB so that, according to this procedure, the wavefunction is written as... 

This represents the total field-free Hamiltonian for a diatomic molecule, to order 1/c2 in the purely electronic terms and to order 1 /Mac2 in the nuclear terms, in a space-fixed axis system of arbitrary origin. We showed in chapter 2 that the solution of a Hamiltonian... 

In conclusion we summarise the total Hamiltonian (excluding nuclear spin effects), written in a molecule-fixed rotating coordinate system with origin at the nuclear centre of mass, for a diatomic molecule with electron spin quantised in the molecular axis system. We number the terms sequentially, and then describe their physical significance. The Hamiltonian is as follows ... 

In this chapter we introduce and derive the effective Hamiltonian for a diatomic molecule. The effective Hamiltonian operates only within the levels (rotational, spin and hyperfine) of a single vibrational level of the particular electronic state of interest. It is derived from the Ml Hamiltonian described in the previous chapters by absorbing the effects of off-diagonal matrix elements, which link the vibronic level of interest to other vibrational and electronic states, by a perturbation procedure. It has the same eigenvalues as the Ml Hamiltonian, at least to within some prescribed accuracy. 

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