Sum-over states theory

A second method is to use a perturbation theory expansion. This is formulated as a sum-over-states algorithm (SOS). This can be done for correlated wave functions and has only a modest CPU time requirement. The random-phase approximation is a time-dependent extension of this method. 

The tensor is given in lower-triangular format (i.e. a. standard orientation. The Approx polarizability line gives the results of the cruder polarizability estimate using sum-over-states perturbation theory, which is suggested by some older texts. 

Malkin, V. G., Malkina, 0. L., Casida, M. E., Salahub, D. R., 1994, Nuclear Magnetic Resonance Shielding Tensors Calculated With a Sum-Over-States Density Functional Perturbation Theory , J. Am. Chem. Soc., 116, 5898. 

The various response tensors are identified as terms in these series and are calculated using numerical derivatives of the energy. This method is easily implemented at any level of theory. Analytic derivative methods have been implemented using self-consistent-field (SCF) methods for a, ft and y, using multiconfiguration SCF (MCSCF) methods for ft and using second-order perturbation theory (MP2) for y". The response properties can also be determined in terms of sum-over-states formulation, which is derived from a perturbation theory treatment of the field operator — [iE, which in the static limit is equivalent to the results obtained by SCF finite field or analytic derivative methods. 

Density functional theory (DFT) calculations employing sum-over-states DF perturbation theory were applied to calculate both the H and chemical shifts of 1,3-dioxane, 1,3-oxathiane, 1,3-dithiane, and the parent cyclohexane <1997JMT(418)231>. Both normal and anomalous trends in the H chemical shifts could be reproduced well and. 

There has been much recent progress in the application of density functional theory (DFT) to the calculation of shift tensors, and several methods are presently available. The sum-over-state (SOS) DFT method developed by Malkin et al. (70) does not explicitly include the current density, but it has been parametrized to improve numerical accuracy. Ziegler and coworkers have described a GIAO-DFT method (71) that is available as part of the Amsterdam density functional package (72). An alternate method developed by Cheeseman and co-workers (73) is implemented in Gaussian 94 (74). 

The onset of sudden variations in vibrational fine structure is one of the most sensitive indicators of a change in resonance structure. The magnitudes of fine-structure parameters are determined by second-order perturbation theory (a Van Vleck or contact transformation) [17]. The energy denominators in these second-order sums over states are approximately independent of vib as long as the <01 <02 - 3/v-6 resonance structure is conserved. 

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. 

Most numerical methods for calculating molecular hyperpolarizability use sum over states expressions in either a time-dependent (explicitly including field dependent dispersion terms) or time-independent perturbation theory framework [13,14]. Sum over states methods require an ability to determine the excited states of the system reliably. This can become computationally demanding, especially for high order hyperpolarizabilities [15]. An alternative strategy adds a finite electric field term to the hamiltonian and computes the hyperpolarizability from the derivatives of the field dependent molecular dipole moment. Finite-field calculations use the ground state wave function only and include the influence of the field in a self-consistent manner [16]. 

A DFT-based third order perturbation theory approach includes the FC term by FPT. Based on the perturbed nonrelativistic Kohn-Sham orbitals spin polarized by the FC operator, a sum over states treatment (SOS-DFPT) calculates the spin orbit corrections (35-37). This approach, in contrast to that of Nakatsuji et al., includes both electron correlation and local origins in the calculations of spin orbit effects on chemical shifts. In contrast to these approaches that employed the finite perturbation method the SO corrections to NMR properties can be calculated analytically from... 

The eigenvectors Ffc determine the oscillator strengths of the excitations. This can be established using the sum-over-states representation of the standard many-electron theory for the dynamic dipole polarizability aav(co) [33]... 

For an elaborated analysis of the relations between structure and hyperpolarizabilities, one has to start from the electronic wavefunctions of a molecule. By using time-dependent perturbation theory, sum-over-states expressions can be derived for the first and second-order hyperpolarizabilities j3 and y. For / , a two-level model that includes the ground and one excited state has proven to be sufficient. For y the situation is more complicated. 

For the evaluation of probabilities for spin-forbidden electric dipole transitions, the length form is appropriate. The velocity form can be made equivalent by adding spin-dependent terms to the momentum operator. A sum-over-states expansion is slowly convergent and ought to be avoided, if possible. Variational perturbation theory and the use of spin-orbit Cl expansions are conventional alternatives to elegant and more recent response theory approaches. 

Uncoupled methods [sometimes called the sum over states (SOS) methods] do not include the field in the Hamiltonian but use a time-dependent perturbation theory approach.38-56 A sum over all excited states is used that requires values for dipole moments in ground and excited states and excitation energies to be evaluated. One must choose the number of states at which to terminate the series. It has been shown in several studies of second-order nonlinearities38 that the /8 values converge after a finite number of states are chosen. Furthermore, this approach intrinsically accounts for frequency dependence. 

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