Sum-over states method

The most recognizable expressions for obtaining nonlinear optical properties are the expressions derived from time-dependent perturbation theory. This procedure is straightforward and has been described in detail. The following equations show the expressions for a, P, and 7  

These equations are difficult to use to accurately calculate the nonlinear optical properties of molecules because of the need for a complete set of excitation energies and transition moments. For p and 7, the transition moments needed are not only ground to excited state (i.e., normal UV/vis-type data) but also excited state to excited state transtion moments. Alternatively, what is needed is a complete description of the ground and excited states of the molecule. 

In practice, the summations over excited states used in the sum-over-states (SOS) expressions generated from quantum mechanical calculations must be truncated in some manner. This leads to two problems (1) how to logically truncate summations and (2) how to ensure that the results have converged such that additional excited states will not change the value. SOS methods are most often used with semiempirical Hamiltonians, where the small basis sets help limit the number of states, precisely because of the problem of logically truncating summations. 

A common truncation is to use single excitation configuration interaction (SCI) results for p and single and double excitation Cl (SDCI) results for 7. The 

One very important simplification based on the SOS formulation is the simple two-state model for static first hyperpolarizabilities and polarizabilities given by  

This expression is general and we can therefore use it together with, e.g. the SCF or MP2 density matrices derived in Section 9.8. 

Turning now to the first-order correction to the field-dependent expectation value ( o(-A) I d I of this operator, we can analogously obtain an expression in 

The sum-over-states method for the calculation of second-or higher-order properties is based on equations like (3.33), (3.110), (3.114) or (3.125), to name a few. The main 

In Section 9.2 it was mentioned that the simplest approximation for an excited state 4 ° ) is to represent it by one singly excited determinant Approximating at the same time the groimd-state wavefunction with the Hartree-Fock determinant 0 and the Hamiltonian by the Hartree-Fock Hamiltonian F, Eq. (9.15), the excitation energies En — E become equal to orbital energy differences ta — and the transition moments (4 q° O 4 ° ) become simple matrix elements of the corresponding one-electron operator d in the molecular orbital basis ( / d ) [see Exercise 10.ll. The spectral representation of the polarization propagator, Eq. (3.110), thus becomes approximated as 

In the static limit, a = 0, this is called the uncoupled Hartree-Fock approximation (UCHF) (Dalgarno, 1959), which played an important role in the early days of calculations of molecular properties. 

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Our theoretical understanding of third-order optical nonlinearity at the microscopic level is really in its infancy. Currently no theoretical method exists which can be reliably used to predict, with reasonable computational time, molecular and polymeric structures with enhanced optical nonlinearities. The two important approaches used are the derivative method and the sum-over-states method (7,24). The derivative method is based on the power expansion of the dipole moment or energy given by Equations 3 and 4. The third-order nonlinear coefficient Y is, therefore, simply given by the fourth derivative of the energy or the third derivative of the induced dipole moment with respect to the applied field. These... 

The sum-over-states method is based on the perturbation expansion of the Stark energy term in which nonlinearities are introduced as a result of mixing with excited states. For example, the expression for Y(—3u) o),a),a)) which will be responsible for third harmonic generation is given as (25)... 

In this calculation one computes the energies and various expectation values of the dipole operator for various excited states. These terms are then summed to compute Y. If one does an exact calculation, in principle both the derivative and the sum-over-states methods should yield the same result. However, such exact calculations are not possible. The sum-over-states method requires that not only the ground states but all excited state properties be computed as well. For this reason one resorts to semi-empirical calculations and often truncates the sum over all states to include only a few excited states. 

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]. 

As an alternative to sum-over-states methods, the perturbation equations can be solved directly. In the context of spin-orbit coupling, reviews on this subject have recently been given by Yarkony117 and by Agren et al.118... 

Four frequently used conventions exist for the definition of non-linear optical polarizabilities, leading to confusion in the realm of NLO. This has been largely clarified by Willets et al. (1992) and in their nomenclature we have used the Taylor series expansion (T convention), originally introduced by Buckingham (1967), where the factorials n are explicitly written in the expansion. Here the polarizabilities of one order all extrapolate to the same value for the static limit w— 0. /3 values in the second convention, the perturbation series (B), have to be multiplied by a factor of 2 to be converted into T values. This is the convention used most in computations following the sum-over-states method (see p. 136). The third convention (B ) is used by some authors in EFISHG experiments and is converted into the T convention by multiplication by a factor of 6. The fourth phenomenological convention (X) is converted to the T convention by multiplication by a factor of 4. 

A- — t dijOab (10.117) P 1 = (T-o pt r ) tions. Modem implementations of propagator methods are computationally related to the derivative techniques discussed in Section 10.3. The significance is that propagator methods allow a calculation of a property directly, without having to construct all the excited states explicitly, i.e. avoiding the Sum Over States method. This also means that... 

De Dominies L, Fantoni R (2006) Effects of electrons statistic on carbon nanotuhes hyperpolarizability frequency dependence determined with sum over states method. J Raman Spectrosc 37 669... 

In the studies of the nonlinearities of organic molecules the sum over states method has played a leading part, to some extent in default of any other practicable approach. 

Keywords First-order hyperpolarizability second-order hyperpolarizability sum-over-states method ... 

Tomonari, M., Ookubo, N., Takada, T. Missing-orbital analysis of molecular hyperpolarizabiUty j8 calculated by a simplified sum-over-states method. Enhancement of the off-diagonal component... 

The sum-over-states method for calculating the resonant enhancement begins with an expression for the resonance Raman intensity, I j-, for the transition from initial state i to final state/in the ground electronic state, and is given by [M]... 

Time-Dependent Perturbation Theory The Sum over States Method. - The... 

These results show that the presence of the solvent leads to an increase of the (a) values with respect to the vacuum, and it reverses its frequency dependence in aqueous solution, unlike in vacuo, the molecule shows a static (a) value which is larger than that at oj 0. This effect can be related to the shape of the e(w) function. By applying Debye formula it is easy to see that, at the frequency considered in our calculations, the value of e(w) is practically equal to e(oo). which is by far smaller than e(0). K one considers the sum-over-states method for the calculation of polarizabilities with these values in minds, then it is easy to give a qualitative explanation of the behavior indicated above. In fact, when the solvent response function is described by e(oo) the actual stabilization of the excited states will be less than in a situation where the same response depends on c(0), hence smaller it will be the correspondent (a) value. ... 

Semiempirical molecular orbital methods have long been a mainstay for the prediction of NLO properties, particularly with sum-over-states methods, and much of our understanding of how molecular properties affect NLO prop-... 

The Sum over States method (SOS) converges slowly i.e., a huge number of states have to be taken into account, including those belonging to a continuum. 

However, the theory for the interaction of matter with the electromagnetic field has to be coherent. The finite field method, so gloriously successful in electric field effects, is in the stone age stage for magnetic field effects. The propagator methods look the most promising, these allow for easier calculation of NMR parameters than the sum-over-states methods. 

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