Zero-order energy eigenvalue

The mechanism of the TP transition, which is principally a second-order process, involves an ED transition from the initial state to an intermediate state (with or without resonance), and thence another from the intermediate state to the final state [221]. However, the states and energies in Eqs. (50)-(52) are only eigenstates and eigenvalues of the zero-order Hamiltonian H0 in a stationary perturbation treatment, in which the complete Hamiltonian is... 

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 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 left and right transition moments of Eqs. (4.15a, 4.15b) determine the solute-solvent interaction in the linear response excitation energies >/ (4.11). This it can be shown by a perturbative analysis of Eq.(4.11). Let us consider as zero-order solutions the eigenvalues and eigenvector and corresponding to excited-states in the presence of the fixed reaction field of the coupled-cluster ground state. At the first order in the solvent perturbation, we can write the excitation energies as... 

Energy corrections are calculated in MS theories as eigenvalues of an effective Hamiltonian, obtained relying on the Bloch equation and the zero-order operator. Matrix elements for the MS-pLMCPT variant at order 2 read as... 

These are two hydrogen-like Schrodinger equations. The eigenvalues E and 2 are hydrogen-like orbital energies. The total electronic energy in the zero-order approximation is... 

The energy eigenvalue for our zero-order ground-state wave function is the sum of two hydrogen-like orbital energies for = 1 ... 

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