Sum-over-states expression

A sum-over-states expression for the coefficient A for the expansion of the diagonal components faaaa was derived by Bishop and De Kee [20] and calculations were reported for the atoms H and He. However, the usual approach to calculate dispersion coefficients for many-electron systems by means of ab initio response methods is still to extract these coefficients from a polynomial fit to pointwise calculated frequency-dependent hyperpolarizabiiities. Despite the inefficiency and the numerical difficulties of such an approach [16,21], no ab initio implementation has yet been reported for analytic dispersion coefficients for frequency-dependent second hyperpolarizabiiities which is applicable to many-electron systems. 

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]. 

The mixed electric dipole-magnetic dipole polarizability G ttiB (co) introduced in Equation (2.98) can be written as a sum-over-states expression [29]... 

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. 

The above consideration influence the complex perturbation expression of the second-order hyperpolarizability y. The number of involved terms can be reduced, leading to a sum-over-states expression of only few terms of interest. 

For a molecule without symmetry reduction the sum-over-states expression for the hyperpolarizability yean often be modeled within the three level approximation (the ground state (index 0) and two excited states (indices 1,2 - not necessarily indicating the hierarchy of the energy eigenvalues)). If we assume po2 Poll f°r substituted molecules with a centrosymmetric backbone as described above, the number of terms reduces leaving only three terms to consider the negative term N > the dipolar term D , and the two-photon term TP, which includes the second excited state 2Ag. In the static limit one obtains for a one-dimensionally conjugated molecule... 

The second-order hyperpolarizability of the tetrasubstituted TEEs are analyzed with the three-level model of the sum-over-states expression. The rotational average of the second-order hyperpolarizability y (Eq.(49)) in the case of THG of a two-dimensional, planar molecule is... 

The increase of second-order hyperpolarizabilities upon backbone elongation has also been evaluated by quantum chemical means by the Bredas group [74]. With a valence effective Hamiltonian approach (VEH/SOS) the parameters in the sum-over-states expression are evaluated leading to the second-order hyperpolarizabilities yof the molecules. With the VEH/SOS approach the description of larger molecules is feasible, which means in the case of PTA molecules longer than the tetramer. 

Note that the formula for a t L xx 0), Eq. (1-151), can be rewritten as the following sum-over-states expression,... 

In resonant infrared multidimensional spectroscopies the excitation pulses couple directly to the transition dipoles. The lowest order possible technique in noncentrosymmetrical media involves three-pulses, and is, in general, three dimensional (Fig. 1A). Simulating the signal requires calculation of the third-order response function. In a small molecule this can be done by applying the sum-over-states expressions (see Appendix A), taking into account all possible Liouville space pathways described by the Feynman diagrams shown in Fig. IB. The third-order response of coupled anharmonic vibrations depends on the complete set of one- and two-exciton states coupled to thermal bath (18), and the sum-over-states approach rapidly becomes computationally more expensive as the molecule size is increased. 

Quantum mechanically, resonance Raman cross-sections can be calculated by the following sum-over-states expression derived from second-order perturbation theory within the adiabatic, Born-Oppenheimer and harmonic approximations... 

The sum-over-states expressions that we have presented in Eqs. (69), (70), and (71) are only true for exact wave functions and they are rather cumbersome methods for calculating time-dependent electromagnetic properties of a quantum mechanical subsystem within a structured environment. The advantage of the sum-overstates expressions is that they illustrate the type of information that is obtainable from response functions. We have utilized modem versions of response theory where the summation over states is eliminated when performing actual calculations, that involve approximative wave functions [21,24,45-47,80-83]. 

The values of w, at which the matrix fl — w2I becomes singular, are very useful because they can be combined with the sum-over-states expression for the mean dynamic polarizability ... 

For the two terms in the sum-over-states expression in Eq. (42) that involve the ground state = 0, the transition frequency a) o is zero. The two terms are of opposite sign and will therefore cancel, and it is common practice to exclude the ground state from the summation and to use a primed summation symbol for the sum over excited states. 

As pointed out above, in the time-dependent case the correlation treatment cannot be based on time-dependent Hartree-Fock orbitals - at least not on the real frequency axis in the vicinity of poles of the response functions. Thus, the polarization of the wavefunction must be described through the variables of the correlation method, i.e. for the CC approach by means of the cluster amplitudes. This has important implications on the choice or suitability of correlation methods. As it is apparent from the sum-over-states expression for the -th response function [96]... 

The sum-over-states expression for the electronic first-order hyperpolarizability (/3) can be written as ... 

By comparison with sum-over-state expressions for the exact case it is possible to identify uansition moments and excited state properties from different residues of the response functions. This is in particular valuable for DFT where it is difficult to straightforwardly extend the theory to excited states and where we have no explicit representation of the excited state wave function. 

The linear pseudoscalar G is tlie isotropic part of the optical rotation tensor, and G = (G -H G, , -H GL)/3. Time-dependent perturbation may be used to obtain a sum-over-states expression for G away from resonance [2, 7] ... 

Except for a factor of 3, the last part of Eq. (7) is readily recognized as the sum-over-states expression for the isotropic invariant of the energy-dependent polarizability of state 0>, i.e. 

To apply these equations, we need the wavefuncdons m> in order to get the dipole moment transition elements and the frequencies spectral series, where only the ground state need be near-exact. This is done by diagonalizing the Hamiltonian matrix formed from a large number of basis functions (which implicitly include the interelectronic coordinate and thus electron correlation). We do this for each symmetry state that is involved. All the ensuing eigenvalues and eigenvectors are then used in the sum-over-states expressions. For helium we require S, P, and D states and for H2 (or D2) E, II, and A states. 

Sum-over-states expression The time-independent second-order properties, such as the nuclear spin-spin coupling constant (see Eq. 13), can be expressed by means of the perturbation theory as... 

Nevertheless, it should be noted that the sum-over-states expression is sometimes also used for the computation of shieldings. It forms the basis, for example, for the so-called propagator ansatze [76, 77] and has been invoked in some empirical schemes for the calculation of shieldings within the DFT framework [78, 79]. 

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