Hamiltonian second-order perturbation theory

The perturbations in this case are between a singlet and a triplet state. The perturbation Hamiltonian, H, of the second-order perturbation theory is spin-orbital coupling, which has the effect of mixing singlet and triplet states. 

This is a simplified Hamiltonian that ignores the direct interaction of any nuclear spins with the applied field, B. Because of the larger coupling, Ah to most transition metal nuclei, however, it is often necessary to use second-order perturbation theory to accurately determine the isotropic parameters g and A. Consider, for example, the ESR spectrum of vanadium(iv) in acidic aqueous solution (Figure 3.1), where the species is [V0(H20)5]2+. 

Second-order Perturbation Theory Treatment of Spin Hamiltonian with Non-coincident... 

Empirically corrected DFT theories almost invariably go back to second-order perturbation theory with expansion of the interaction Hamiltonian in inverse powers of the intermolecular distance, leading to R 6, R x, and R 10 corrections to the energy in an isotropic treatment (odd powers appear if anisotropy is taken into account [86]). 

Gv( f) covering symmetry67. For orientations of B0 in the mirror plane S, the symmetry group of the spin Hamiltonian is < 9f = C2h(e2f). The direct product base of the nuclear spin functions of two geometrically equivalent nuclei reduces to two classes, containing six A-type and three B-type functions, respectively. Second order perturbation theory applied to H = UtHU, where U symmetrizes the base functions of the Hamil-... 

In the vicinity of the atomic absorption edges, the participation of free and bound excited states in the scattering process can no longer be ignored. The first term in the interaction Hamiltonian of Eq. (1.11) leads, in second-order perturbation theory, to a resonance scattering contribution (in units of classical electron scattering) equal to (Gerward et al. 1979, Blume 1994)4... 

The total binding energy of a NFE metal can be evaluated within second-order perturbation theory. In the presence of a perturbation fi to the Hamiltonian operator of a system, the energy of state is given by... 

A more widely used approach for organic molecules is based on second-order perturbation theory. Here the dipolar contribution to the field induced charge displacement is calculated by inclusion of the optical field as a perturbation to the Hamiltonian. Since the time dependence of the field is included here, dispersion effects can be accounted for. In this approach the effect of the external field is to mix excited state character into the ground state leading to charge displacement and polarization. The accuracy of this method depends on the parameterization of the Hamiltonian in the semi-empirical case, the extent to which contributions from various excited states are incorporated into the calculation, and the accuracy with which those excited states are described. This in turn depends on the nature of the basis set and the extent to which configuration interaction is employed. This method is generally referred to as the sum over states (SOS) method. 

The above approaches estimate the excitation rate by using either second-order perturbation theory [6] or a re-summation to all orders in perturbation theory [20,21]. In order to be able to sum the infinite series of perturbation theory references [20,21], we use an orthogonal basis-set of the model Hamiltonian (2) (the creation and destruction operators need to... 

A similar interaction would be observed between all Fermi polyads containing sets of vibrational levels related by the selection rule A tv = 2, A tv = +1, and the hamiltonian matrix should be diagonalized for each Fermi polyad without the use of perturbation theory. If, on the other hand, the interaction (63) were smaller, or the separation between the unperturbed levels were larger, the interaction could be treated by perturbation theory it can be shown that, in second-order perturbation theory, equation (63) would contribute a term to the vibrational anharmonic constants... 

The application of second-order perturbation theory to the Hiickel-Hubbard Hamiltonian for z close to one (Section 4.2) yields the freeon, antiferromagnetic, Heisenberg exchange Hamiltonian,... 

The Heisenberg Hamiltonians are derived by the application of second-order perturbation theory to the extended Huckel-Hubbard Hamiltonian at z close to -l(ferromagnet) and at z close to + 1 (antiferromagnet) respectively. 

RELATIVISTIC MULTIREFERENCE PERTURBATION THEORY COMPLETE ACTIVE-SPACE SECOND-ORDER PERTURBATION THEORY (CASPT2) WITH THE FOUR-COMPONENT DIRAC HAMILTONIAN... 

Using second-order perturbation theory and evaluating the matrix elements of the vibrational and rotational operators occuring in our simplified Hamiltonian... 

Andersson, K. Different forms of the zeroth-order Hamiltonian in second-order perturbation theory with a complete active space self-consistent field reference function, Theor. Chim. Acta 1995, 91, 31-46. 

The virtual orbitals / are optimised using a second-order perturbation theory approximation to the energy [31] so that we need only evaluate the diagonal and first row elements of the hamiltonian and overlap matrices ... 

Crystal states, where one molecule is ionized, and the conduction band contains one electron, can be used as those intermediate states, as has been shown in (31) (the role of such intermediate states in theory of photoconductivity of molecular crystals has been discussed by Lyons (32)). The use of intermediate states becomes indispensable when the second-order perturbation theory is applied in the case of a degenerate term. According to (33), correct linear combinations of crystal states, containing one molecule in a triplet state and all remaining in the ground state, can be found by perturbation theory when in the corresponding secular equation the following effective Hamiltonian is used... 

A first orbital-dependent approximation for the relativistic Ec has been derived via second order perturbation theory with respect to the relativistic KS Hamiltonian [54], Within the no-pair approximation and neglecting the transverse interaction this second order term reads... 

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