First-order correction to energy

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-... 

These are zero-, first-, second-, th-order perturbation equations. The zero-order equation is just the Schodinger equation for the unperturbed problem. The first-order equation contains two unknowns, the first-order correction to the energy, Wi, and the first-order correction to the wave function, 4< i. The th-order energy correction can be calculated by multiplying from the left by 4>o and Integrating, and using the turnover rule ( o Ho, ) = (, Ho o)... 

Up to this point we are still dealing with undetermined quantities, energy and wave funetion corrections at each order. The first-order equation is one equation with two unknowns. Since the solutions to the unperturbed Schrddinger equation generates a complete set of functions, the unknown first-order correction to the wave function can be expanded in these functions. This is known as Rayleigh-Schrddinger perturbation theory, and the equation in (4.32) becomes... 

The expansion coefficients determine the first-order correction to the perturbed wave function (eq. (4.35)), and they can be calculated for the known unperturbed wave functions and energies. The coefficient in front of 4>o for 4 i cannot be determined from the above formula, but the assumption of intermediate normalization (eq. (4.30)) makes Co = 0. 

The formula for the first-order correction to the wave function (eq. (4.37)) similarly only contains contributions from doubly excited determinants. Since knowledge of the first-order wave function allows calculation of the energy up to third order (In - - 1 = 3, eq. (4.34)), it is immediately clear that the third-order energy also only contains contributions from doubly excited determinants. Qualitatively speaking, the MP2 contribution describes the correlation between pairs of electrons while MP3 describes the interaction between pairs. The formula for calculating this contribution is somewhat... 

In other words, the diagonal elements of the perturbing Hamiltonian provide the first-order correction to the energies of the spin manifold, and the nondiagonal elements give the second-order corrections. Perturbation theory also provides expressions for the calculation of the coefficients of the second-order corrected wavefunctions l / in terms of the original wavefunctions (p)... 

In order to get the first-order correction to the energy E (2) is premultiplied by ip k and integrated over all space, giving... 

In the previous section several equations were described that can be used to calculate MCD spectra. If the spectra are to be calculated using the transition-based approach described in Sections II.A.1-II.A.4, a number of quantities must be evaluated. These include the perturbed and unperturbed excitation energies, the perturbed and unperturbed transition moments between the ground and excited states, and/or the magnetic moment of the ground state. If an MCD spectrum is to be calculated with the imaginary Verdet approach described in Section II.A.6, then the first-order correction to the frequency-dependent polarizability due to a magnetic field is required. 

Equation (40) is a generalized eigenvalue equation. The eigenvalue j is interpreted as the excitation energy from the ground state to the Jth excited state. The vectors Xj and Yj are the first-order correction to the density matrix at an excitation and describe the transition density between the ground state and the excited state J. 

Since (°) is a Hermitian operator and 4 is an eigenfunction of it, the first and third integrals are equal and cancel, leaving an expression for the first-order correction to the energy,... 

It is generally believed that a correction to the energy which is comparable to the first-order correction to the wave function would involve the second-order term En which may be extracted from the second-order equation (A.98). Multiply every term on the left by and integrate ... 

The first-order correction to the energy of the state with quantum numbers vJM is [Equation (1.201)]... 

MP2 (note that with the partitioning of Eq. 2.3 there is no first-order correction to the energy), is probably the cheapest available treatment of dynamical correlation. 

The first of these is already solved, by assumption. If the second can be solved, we can find the first-order corrections to the wavefunction and the energy Solution of the third equation gives the second-order corrections, and so on. It is shown in the standard textbooks (e.g. Eyring, Walter and Kimball, 1944) that the solutions are... 

In the identification of W 3) we have used the first-order response equations [Eq. (64)] to eliminate A(2). From Eqs. (68) and (69) we see that the first-order correction to the wave function determines the energy through third order. In general the nth-order response of the wave function determines the energy through order 2n + 1. 

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