Hamiltonian energy operator

B. Now operate on the wavefunctions with an operator P, which may be a spin operator pn n) = 23 j cn]P j pj), here pn is the corresponding eigenvalue. If the operator is an energy (Hamiltonian) operator, then the eigenvalue is energy. 

In this scheme, the INT is partitioned into electrostatic, exchange, repulsion, polarization, and dispersion. Let us now look at the mathematical derivation of the various components. For a molecular system (AB) with wavefunction O and total energy Hamiltonian operator H, the expectation value is given as... 

The form (3.10) allows introducing the total energy (Hamiltonian) operator of the system... 

Applications of quantum mechanics to chemistry invariably deal with systems (atoms and molecules) that contain more than one particle. Apart from the hydrogen atom, the stationary-state energies caimot be calculated exactly, and compromises must be made in order to estimate them. Perhaps the most useful and widely used approximation in chemistry is the independent-particle approximation, which can take several fomis. Conuiion to all of these is the assumption that the Hamiltonian operator for a system consisting of n particles is approximated by tlie sum... 

There is a very convenient way of writing the Hamiltonian operator for atomic and molecular systems. One simply writes a kinetic energy part — for each election and a Coulombic potential Z/r for each interparticle electrostatic interaction. In the Coulombic potential Z is the charge and r is the interparticle distance. The temi Z/r is also an operator signifying multiply by Z r . The sign is - - for repulsion and — for atPaction. 

The sum of two operators is an operator. Thus the Hamiltonian operator for the hydrogen atom has — j as the kinetic energy part owing to its single election plus — 1/r as the electiostatic potential energy part, because the charge on the nucleus is Z = 1, the force is atrtactive, and there is one election at a distance r from the nucleus... 

This equation is an eigenvalue equation for the energy or Hamiltonian operator its eigenvalues provide the energy levels of the system... 

Once a wave function has been determined, any property of the individual molecule can be determined. This is done by taking the expectation value of the operator for that property, denoted with angled brackets < >. For example, the energy is the expectation value of the Hamiltonian operator given by... 

The Hamiltonian operator is the quantum-mechanical analogue of the energy, and we say that the allowed values of the energy, the , above, are the eigenvalues... 

The spin Hamiltonian operates only on spin wavefunctions, and all details of the electronic wavefunction are absorbed into the coupling constant a. If we treat the Fermi contact term as a perturbation on the wavefunction theR use of standard perturbation theory gives a first-order energy... 

One may attempt to approximate to such an experimental situation by considering a subsystem with small dimensions in the direction of the flow, so that a single temperature may be sufficiently precise in describing it. In this model one would have to provide a time-dependent hamiltonian operating in such a way as to feed energy into the system at one boundary and to remove energy from the other boundary. We would therefore be obliged to discuss systems with hamiltonians that are explicitly functions of time, and also located on the boundaries of the macrosystem. 

Hamiltonian operator, 2,4 for many-electron systems, 27 for many valence electron molecules, 8 semi-empirical parametrization of, 18-22 for Sn2 reactions, 61-62 for solution reactions, 57, 83-86 for transition states, 92 Hammond, and linear free energy relationships, 95... 

Now consider a d ion as an example of a so-called many-electron atom. Here, each electron possesses kinetic energy, is attracted to the (shielded) nucleus and is repelled by the other electron. We write the Hamiltonian operator for this as follows ... 

The one-electron Hamiltonian operator h = h +v, with kinetic energy operator, generates a complete spectrum of orbitals according to the Schrodinger equation... 

We show next that the parity operator IT commutes with the Hamiltonian operator H if the potential energy F(q) is an even function of q. The kinetic energy term in the Hamiltonian operator is given by... 

A useful expression for evaluating expectation values is known as the Hell-mann-Feynman theorem. This theorem is based on the observation that the Hamiltonian operator for a system depends on at least one parameter X, which can be considered for mathematical purposes to be a continuous variable. For example, depending on the particular system, this parameter X may be the mass of an electron or a nucleus, the electronic charge, the nuclear charge parameter Z, a constant in the potential energy, a quantum number, or even Planck s constant. The eigenfunctions and eigenvalues of H X) also depend on this... 

Since the Hamiltonian operator is hermitian, the energy eigenvalues E are real. 

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