Hyperpolarizabilities wavefunctions

The CCSD model gives for static and frequency-dependent hyperpolarizabilities usually results close to the experimental values, provided that the effects of vibrational averaging and the pure vibrational contributions have been accounted for. Zero point vibrational corrections for the static and the electric field induced second harmonic generation (ESHG) hyperpolarizability of methane have recently been calculated by Bishop and Sauer using SCF and MCSCF wavefunctions [51]. 

This calculation has shown the importance of the basis set and in particular the polarization functions necessary in such computations. We have studied this problem through the calculation of the static polarizability and even hyperpolarizability. The very good results of the hyperpolarizabilities obtained for various systems give proof of the ability of our approach based on suitable polarization functions derived from an hydrogenic model. Field—induced polarization functions have been constructed from the first- and second-order perturbed hydrogenic wavefunctions in which the exponent is determined by optimization with the maximum polarizability criterion. We have demonstrated the necessity of describing the wavefunction the best we can, so that the polarization functions participate solely in the calculation of polarizabilities or hyperpolarizabilities. 

The present paper is aimed at developing an efficient CHF procedure [6-11] for the entire set of electric polarizabilities and hyperpolarizabilities defined in eqs. (l)-(6) up to the 5-th rank. Owing to the 2n+ theorem of perturbation theoiy [36], only 2-nd order perturbed wavefunctions and density matrices need to be calculated. Explicit expressions for the perturbed energy up to the 4-th order are given in Sec. IV. 

A computer program for the theoretical determination of electric polarizabilities and hyperpolarizabilitieshas been implemented at the ab initio level using a computational scheme based on CHF perturbation theory [7-11]. Zero-order SCF, and first-and second-order CHF equations are solved to obtain the corresponding perturbed wavefunctions and density matrices, exploiting the entire molecular symmetry to reduce the number of matrix element which are to be stored in, and processed by, computer. Then a /j, and iap-iS tensors are evaluated. This method has been applied to evaluate the second hyperpolarizability of benzene using extended basis sets of Gaussian functions, see Sec. VI. 

Our present focus is on correlated electronic structure methods for describing molecular systems interacting with a structured environment where the electronic wavefunction for the molecule is given by a multiconfigurational self-consistent field wavefunction. Using the MCSCF structured environment response method it is possible to determine molecular properties such as (i) frequency-dependent polarizabilities, (ii) excitation and deexcitation energies, (iii) transition moments, (iv) two-photon matrix elements, (v) frequency-dependent first hyperpolarizability tensors, (vi) frequency-dependent polarizabilities of excited states, (vii) frequency-dependent second hyperpolarizabilities (y), (viii) three-photon absorptions, and (ix) two-photon absorption between excited states. 

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. 

In (II) the reaction field of the dipole is included in the molecular Hamiltonian, so that the QM calculation, at whatever level, is modified to give a new molecular wavefunction for one molecule at the centre of a cavity. This calculation can be carried out in the absence of an applied macroscopic field and would give the unperturbed properties (dipole moment, energy states etc) of a solvated molecule. The macroscopic field has then usually been applied in a finite field calculation of the hyperpolarizability. One source of uncertainty in this procedure arises from the fact that when the reaction field is introduced into the hamiltonian it appears in a specific form,... 

The difference equation or numerical integration method for vibrational wavefunctions usually referred to as the Numerov-Cooley method [111] has been extended by Dykstra and Malik [116] to an open-ended method for the analytical differentiation of the vibrational Schrodinger equation of a diatomic. This is particularly important for high-order derivatives (i.e., hyperpolarizabilities) where numerical difficulties may limit the use of finite-field treatments. As in Numerov-Cooley, this is a procedure that invokes the Born-Oppenheimer approximation. The accuracy of the results are limited only by the quality of the electronic wavefunction s description of the stretching potential and of the electrical property functions and by the adequacy of the Born-Oppenheimer approximation. 

For nonlinear (magneto-) optical properties, calculations of an accuracy close to that of modern gas phase experiments require - similar to what has also been found for other properties like structures [79, 109], reaction enthalpies [79, 110, 111], vibrational frequencies [112, 113], NMR chemical shifts [114], etc. - at least an approximate inclusion of connected triple excitations in the wavefunction. This has been known for years now from calculations of static hyperpolarizabilities with the CCSD(T) approximation [9-13]. CCSD(T) accounts rather efficiently for connected triples through a perturbative correction on top of CCSD. For the reasons pointed out in Section 2.1 CCSD(T) is, as a two-step approach, not suitable for the calculation of frequency-dependent properties. Therefore, the CC3 model has been proposed [56, 58] as an alternative to CCSD(T) especially designed for use in connection with response theory. CC3 is an approximation to CCSDT - alike CCSDT-la and related methods - where the triples equations are truncated such that the scaling of the computational efforts with system size is reduced to as for CCSD(T),... 

The situation is somewhat different for the convergence with the wavefunction model, i.e. the treatment of electron correlation. As an anisotropic and nonlinear property the first dipole hyperpolarizability is considerably more sensitive to the correlation treatment than linear dipole polarizabilities. Uncorrelated methods like HF-SCF or CCS yield for /3 results which are for small molecules at most qualitatively correct. Also CC2 is for higher-order properties not accurate enough to allow for detailed quantitative studies. Thus the CCSD model is the lowest level which provides a consistent and accurate treatment of dynamic electron correlation effects for frequency-dependent properties. With the CC3 model which also includes the effects of connected triples the electronic structure problem for j8 seems to be solved with an accuracy that surpasses that of the latest experiments (vide infra). 

Naively, one would expect that second hyperpolarizabilities y are theoretically and experimentally more difficult to obtain than first hyperpolarizabilities (3. From a computational point of view the calculation of fourth-order properties requires, according to the 2n + 1-rule, second-order responses of the wavefunction and thus the solution of considerably more equations than needed for j3 (cf. Section 2.3). However, unlike (3 the second dipole hyperpolarizability y has two isotropic tensor... 

Since second hyperpolarizabilities depend in addition to the first-order also on the second-order response of the wavefunction, the minimal requirements with respect to the choice of basis sets are for y somewhat higher than for the linear polarizabilities a and the first hyperpolarizabilities j8, in particular for atoms and small molecules. For the latter at least doubly-polarized basis sets augmented with a sufficient number of diffuse functions (e.g. d-aug-cc-pVTZ or t-aug-cc-pVTZ) are needed to obtain qualitatively correct results. Highly accurate results at a correlated level will in general only be obtained in quadruple- or better basis sets. 

Hyperpolarizabilities can be calculated in a number of different ways. The quantum chemical calculations may be based on a perturbation approach that directly evaluates sum-over-states (SOS) expressions such as Eq. (14), or on differentiation of the energy or induced moments for which (electric field) perturbed wavefunctions and/or electron densities are explicitly calculated. These techniques may be implemented at different levels of approximation ranging from semi-empirical to density functional methods that account for electron correlation through approximations to the exact exchange-correlation functionals to high-level ab initio calculations which systematically include electron correlation effects. 

To obtain hyperpolarizabilities of calibrational quality, a number of standards must be met. The wavefunctions used must be of the highest quality and include electronic correlation. The frequency dependence of the property must be taken into account from the start and not be simply treated as an ad hoc add-on quantity. Zero-point vibrational averaging coupled with consideration of the Maxwell-Boltzmann distribution of populations amongst the rotational states must also be included. The effects of the electric fields (static and dynamic) on nuclear motion must likewise be brought into play (the results given in this section include these effects, but exactly how will be left until Section 3.2.). All this is obviously a tall order and can (and has) only been achieved for the simplest of species He, H2, and D2. Comparison with dilute gas-phase dc-SHG experiments on H2 and D2 (with the helium theoretical values as the standard) shows the challenge to have been met. 

The foregoing formula refer to changes in the hamiltonian of the system. The polarizability and hyperpolarizability terms arise from the changes in the wavefunction induced by the perturbed hamiltonian. 

The terms that are nonlinear in a arise because the state wavefunction 0> depends on a (i.e., the state has responded to a//, which gives rise to the name response property"). When, for example, a// j represents a static electric field (a//, = 6 r), Ej yields the permanent electric dipole moment (/i) of the unperturbed state 0>, Ej gives this stale s polarizability (oc), and E, E4, etc. yield successively higher hyperpolarizabilities (/i,y, etc.). 

A scheme to calculate frequency-dependent first hyperpolarizabilities for general CC wavefunctions (CCSD, CC3, CCSDT, and CCSDTQ) has been presented by O Neill et al This analytical third derivative scheme exploits the similarities between response theory and analytic derivative theory. Illustrations have first confirmed that the inclusion of higher-than-double excitations is essential for a quantitative description of the first hyperpolarizabilities. Moreover, the CC3 approximation has been seen to provide good results for singly-bonded systems, with little multireference character, but that full triples contribution using CCSDT are required for benchmark quality results on other systems. Representative results of ref. 18 are given in Table 1. 

Among the ab initio wavefunction methods including electron correlation effects, only a few can predict the dynamic hyperpolarizabilities of molecules containing 20-50 atoms. As a consequence, approximate schemes have been developed to estimate the dynamic correlated values from... 

Calculations of atomic and molecular hyperpolarizabilities usually proceed via time-dependent perturbation theory for the perturbed atomic states. Even for molecules of modest size, the calculation of the complete set of unperturbed wavefunctions, and exact calculation of the hyperpolarizabilities, is prohibitively difficult. Liquid crystals typically consist of organic molecules with aromatic cores, and there is considerable experimental [10] and theoretical [11, 12] evidence to indicate that the dominant contribution to the polarizabilities originates from the delocalized r-electrons in conjugated regions of these molecules. Even considering only r-electrons the calculations rapid-... 

The finite-field method can thus be applied at any level of approximation or correlation and even to approximations or methods for which a wavefunction or a ground-state energy is not defined. The latter approach was used for example for the calculation of the static second hyperpolarizability 7(0 0,0,0) of Li as second derivative of a(0 0) at the SOPPA(CCSD) level (Sauer, 1997). 

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