Hamiltonian hydrogen molecule

II. Crude Born-Oppenheimer Approximarion in. Hydrogen Molecule Hamiltonian... 

Tie hydrogen molecule is such a small problem that all of the integrals can be written out in uU. This is rarely the case in molecular orbital calculations. Nevertheless, the same irinciples are used to determine the energy of a polyelectronic molecular system. For an ([-electron system, the Hamiltonian takes the following general form ... 

Sketch the hydrogen molecule system (2 protons and 2 electrons) and verify the Hamiltonian 10.3.1. 

Hydrogen abstraction reactions potential surfaces for, 25-26,26,41 resonance structures for, 24 Hydrogen atom, 2 Hydrogen bonds, 169,184 Hydrogen fluoride, 19-20, 20,22-23 Hydrogen molecules, 15-18 energy of, 11,16,17 Hamiltonian for, 4,15-16 induced dipoles, 75,125 lithium ion effect on, 12... 

Oppenheimer approximation, hydrogen molecule, Hamiltonian equation, 514— 516... 

The hydrogen molecule is the simplest molecule that contains an electron-pair bond. The electronic Hamiltonian for H2, with a term for nuclear repulsion, is... 

From the classical electrodynamics, the Dirac Hamiltonian of a hydrogen molecule moving in a constant magnetic field B is [102]... 

Hence, the Hamiltonian constructed from equations (1) and (5) is suitable for the evaluation of the transverse magnetic effect on a hydrogen molecule, due to the presence of a parallel magnetic field, that gives rise to the fine features of PECs. Since equation (12) commutes with the total Hamiltonian the correction for... 

In principle, it should be possible to obtain the electronic energy levels of the molecules as a solution of the Schrodinger equation, if inter-electronic and internuclear cross-coulombic terms are included in the potential energy for the Hamiltonian. But the equation can be solved only if it can be broken up into equations which are functions of one variable at a time. A simplifying feature is that because of the much larger mass of the nucleus the motion of the electrons can be treated as independent of that of the nucleus. This is known as the Bom-Oppen-heimer approximation. Even with this simplification, the exact solution has been possible for the simplest of molecules, that is, the hydrogen molecule ion, H + only, and with some approximations for the H2 molecule. 

Next we consider the hydrogen molecule. The electronic Hamiltonian of H2 is... 

It is not possible to obtain a direct solution of a Schrodinger equation for a structure containing more than two particles. Solutions are normally obtained by simplifying H by using the Hartree-Fock approximation. This approximation uses the concept of an effective field V to represent the interactions of an electron with all the other electrons in the structure. For example, the Hartree-Fock approximation converts the Hamiltonian operator (5.7) for each electron in the hydrogen molecule to the simpler form ... 

The Hamiltonian (1) is spin free, commutative with the spin operator S2 and its z-component Sz for one-electron and many-electron systems. The total spin operator of the hydrogen molecule relates to the constituent one-electron spin operators as... 

It was Pauling and his collaborators who generalized the Heitler-London treatment of the hydrogen molecule to general polyelectronic systems and gave the rules for evaluating the Hamiltonian matrix elements [3,4], which could also... 

In the discussion of the hydrogen molecule, it was pointed out that there are two energy levels, Ei corresponding to a symmetrical coordinate wave function (singlet spin function) and En corresponding to an antisymmetrical coordinate wave function (triplet spin functions). If the part of the Hamiltonian labeled H in equation 21 is treated as a perturbation, the energies of the singlet and triplet states are... 

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