Eigenvalue second-order correction

The second-order correction E to the eigenvalue E is obtained by multiplying equation (9.23) by and integrating over all space... 

With the first-order eigenvalue and wave function corrections in hand, we can carry out analogous operations to determine the second-order corrections, then the third-order, etc. The algebra is tedious, and we simply note the results for the eigenvalue corrections, namely... 

Second order corrections to the energy of sp3 carbon atom. In order to construct the required mechanistic picture, the estimate of the restoring force which opposes both the quasi- and pseudorotation (deformation) of the hybridization tetrahedra is necessary. That can be obtained by a linear response procedure. For the sp3 carbon atom in the symmetric tetrahedral environment, the related resonance energy is a diagonal quadratic form with respect to small quasi- and pseudorotations together with triply degenerate eigenvalues [44,45] ... 

We also see that, in these cases, the first-order correction t//J to the eigenfunction determines the second-order correction E to the eigenvalue. 

Assuming that the conditions for the application of perturbation theory are obeyed, we arrive at the second-order correction to the eigenvalues... 

Exercise 3.14 Derive the results for the first- and second-order corrections to the eigenvalues using Eqs. (3.187) and (3.188) and the zeroth-order eigenvalue problems Eq. (3.179). 

The first-order correction to the wave function and the second-order correction to the energy eigenvalue are more complicated than the first-order correction to the energy eigenvalue. Appendix E contains the formulas for these quantities. No exact calculation of the second-order correction to the energy of the helium atom has been made, but a calculation made by a combination of the perturbation and variation methods gives an accurate upper bound ... 

Then the sum over all eigenvalues Aj., to second order in the orbital corrections, becomes... 

The second-order terms of the effective Hamiltonian are proportional to the sd parameter to the third power, and in general, an order correction is proportional to The correction terms shift eigenvalues and hence the... 

The fact that the term of order s3 in Eq. (144) is off-block-diagonal implies that if we perform a second unitary transformation ee W3, there will be no term of order s3 in the diagonal block projection Ih, and thus the next order correction for the diagonal block, and therefore for eigenvalues, will be of order 4 (given by... 

One way to derive this is to eliminate the coefficients for the upper set between Eqs. (19-19). This leads to an eigenvalue equation for eigenvalues of the matrix <5o /) -hZj, W y Wyi,. The sum of eigenvalues (sum of E ) over this lower set is exactly equal to the trace of this matrix. Similarly, the sum of E over the set is exactly equal to the trace of the squared matrix. One can also write these sums as sums over , = — ITj H- A,-, noticing that A,- has second-order and fourth-order terms, and solve for the sum of Eq. (19-20). It is also readily confirmed that this is correct to fourth order for the special case of only two coupled states by expanding the exact solution, >/lTf2 + VVj, in Wi2-... 

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