Energy eigenfunctions zero-order

The final state of the system, corresponding to the ground state of the molecule plus a photon, is represented by ipB = 0 k, e>. and the set 0, are the eigenfunctions of Hth while vac ) and k, e) represent the zero-photon and the one-photon eigenstates of HR, respectively. The time evolution of the amplitudes a,(t) and CE(t) can be computed from time-dependent perturbation theory. The equations of motion are determined by the energy levels of the zero-order states of Hel + HR, by the coupling matrix elements... 

The multidimensional eorrelated basis fnnetions for the full molecule are then dehned as energy selected products of eigenfunctions of zero order Hamiltonians. 

The average value of the dipole moment will be calculated by means of Dirac s perturbation theory for nonstationary. states, up to third order the zero order refers to the free molecules in the absence of the field. Let the wave function of the system of the two interacting molecules in- the external field be specified by y, an eigenfunction of the total Hamiltonian H. This wave function y> may be expanded in a complete set of the energy eigenfunctions unperturbed system the index n labels the various unperturbed eigenstates characterized by the energy En. We may then write... 

The choice of zero-order model determines the names used to label the observed energy levels. These names are a conventionally agreed upon (Jenkins, 1953) set of shorthand labels for the basis functions tpj, not for the eigenfunctions ipi- Since the eigenfunctions of the exact H are linear combinations of basis functions... 

The first-order corrections to energy and the zero-order wave functions, with expansion coefficients c, are obtained as the eigenvalues and eigenfunctions. Because the matrix V generally has p different eigenvalues, the degeneracy of the refer-... 

The three dimensional (3-D) basis functions are then defined as products of eigenfunctions of zero-order Hamiltonians with the energy cut-off criterion. The eigenfunctions for each coordinate, are represented in the primitive DVR... 

The ability to assign a group of vibrational/rotational energy levels implies that the complete Hamiltonian for these states is well approximated by a zero-order Hamiltonian which has eigenfunctions /,( i)- The are product functions of a zero-order orthogonal basis for the molecule, or, more precisely, product functions in a natural basis representation of the molecular states, and the quantity m represents the quantum numbers defining tj>,. The wave functions are given by... 

The resolvant can be expressed in terms of the eigenfunctions and eigenvalues of the zero-order problem. Substituting equation (38) into the expressions given above for the perturbed wavefunetions and energies gives the sum-over-states formulae most often used in practical applications of perturbation theory. 

The wave function is one of the energy eigenfunction of the molecule in the absence of radiation. Inclusion of the perturbation produces a time-dependent wave function, which is written as a linear combination of the zero-order wave functions ... 

The zero order wavefunctions, labeled by the quantum numbers and mi, are eigenfunctions of Iz and Sz, (and of and S ) and the associated energy values, represented in Figure 12.10, are... 

Let us therefore suppose that we have set up an approximate potential field V for the molecule, in which an electron is to move. This field might, for example, be that obtained by the superposition of the Hartree fields of the component atoms. Let us also suppose that we are given a set of functions atomic orbitals of the various atoms of the molecule, although any set of independent functions could be used. The approximate orbitals can then be found by the method of trial eigenfunctions, using zero-order functions. The approximate energies of the molecular orbitals will therefore be the roots of the secular equation... 

For o(l) aiid 6 (2) we take the Is eigenfunctions of hydrogen. Using the perturbation above, we immediately see that the first-order perturbation energy is zero, since H is an odd function of the coordinates while the s are even functions of the coordinates. According to equation 7-27, the second-order perturbation energy is... 

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