Frequency-dependent first hyperpolarizability

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. 

In case of the frequency-dependent first hyperpolarizability, we note that the average value of the hyperpolarizability changes sign as a water molecule is transferred from vacuum to the condensed phase. This observation has also been observed experimentally. Furthermore, the effects of the polarization terms in the structured environment are important since the quadratic response calculation within the MCSCF/CM approach without the polarization interactions leads to much smaller values for the average hyperpolarizability. 

Table 1.10 Static and frequency-dependent first hyperpolarizabilities (10 esu) in vacuo and in water.

Theory of Frequency-Dependent First Hyperpolarizability. Calculations in the Dipole Length and Mixed-Velocity Formalisms. 

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. 

One of the hurdles in this field is the plethora of definitions and abbreviations in the next section I will attempt to tackle this problem. There then follows a review of calculations of non-linear-optical properties on small systems (He, H2, D2), where quantum chemistry has had a considerable success and to the degree that the results can be used to calibrate experimental equipment. The next section deals with the increasing number of papers on ab initio calculations of frequency-dependent first and second hyperpolarizabilities. This is followed by a sketch of the effect that electric fields have on the nuclear, as opposed to the electronic, motions in a molecule and which leads, in turn, to the vibrational hyperpolarizabilities (a detailed review of this subject has already been published [2]). Section 3.3. is a brief look at the dispersion formulas which aid in the comparison of hyperpolarizabilities obtained from different processes. 

From a basis set study at the CCSD level for the static hyperpolarizability we concluded in Ref. [45] that the d-aug-cc-pVQZ results for 7o is converged within 1 - 2% to the CCSD basis set limit. The small variations for the A, B and B coefficients between the two triple zeta basis sets and the d-aug-cc-pVQZ basis, listed in Table 4, indicate that also for the first dispersion coefficients the remaining basis set error in d-aug-cc-pVQZ basis is only of the order of 1 - 2%. This corroborates that the results for the frequency-dependent hyperpolarizabilities obtained in Ref. [45] by a combination of the static d-aug-cc-pVQZ hyperpolarizability with dispersion curves calculated using the smaller t-aug-cc-pVTZ basis set are close to the CCSD basis set limit. 

Heterogeneous dielectric media models have included the developments of Jprgensen et al. [7-9] (reviewed here) and Corni and Tomasi [52,53], Generally, the number of methods for determining frequency-dependent molecular electronic properties, such as the polarizability or first- and second hyperpolarizability tensors of heterogeneously solvated molecules, is very limited. 

Working to similar levels of accuracy, Pawlowski et al have calculated the static and frequency-dependent linear polarizability and second hyperpolarizability of the Ne atom using coupled-cluster methods with first order relativistic corrections. Good agreement with recent experimental results is achieved. Klopper et al.s have applied an implementation of the Dalton code that enables... 

Norman and Jensen27 have implemented a method for obtaining second order response functions within the four component (relativistic) time-dependent Hartree-Fock scheme. Results are presented for the first order hyperpolarizabilities for second harmonic generation, />(—2o o),o ) for CsAg and CsAu. A comparison of the results with those of non-relativistic calculations implies that the nonrelativistic results are over-estimated by 18% and 66% respectively. In this method transitions that are weakly-allowed relativistically can lead to divergences in the frequency-dependent response, which would be removed if the finite lifetimes of the excited states could be taken into account. 

The present contribution reviews recent advances in the highly accurate calculation of frequency-dependent properties of atoms and small molecules, electronic struc-mre methods, basis set convergence and extrapolation techniques. Reported applications include first and second hyperpolarizabilities, Faraday, Buckingham and Cotton-Mouton effects as well as Jones and magneto-electric birefringence... 

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). 

The above expressions become slightly modified when the frequency dependent is evaluated. In order to calculate the first-order hyperpolarizability for a second-harmonic generation (SHG) process (/3 (2 u)), the denominator in Eqs. (4) and (5) should be modified by a factor reflecting the frequency dependence (the... 

As I have said, Sekino and Bartlett [31] were the first to show how to proceed to calculate frequency-dependent hyperpolarizabilities within the TDCPHF approximation. They developed an infinite-order recursive procedure, using density matrices, and, by solving the equations iteratively at each order, could, in principle, calculate any non-linear optical property. Their first application was to H2, FH (the work on FH was analysed in detail in another paper [38]), CH4 and the fluoromethanes. The processes SHG, OR, dc-SHG, dc-OR, IDRI and THG were considered but not all hyperpolarizability components were computed (the assumption of Kleinman symmetry was made). 

The PCM calculation are performed by using the CPHF formalism [51] for the static case, and to the TD-CPHF formalism for the frequency dependent case [52]. There are also calculations at higher levels of the QM theory which have not been fully analyzed. The formulas are quite complex, and we refer the interested readers to the two source papers. What is worth remarking here is that (hyper)polarizability values are quite sensitive to the cavity errors. In passing from 7I ) (i.e. a, the polarizability tensor) to 7 1 (i.e. /9, the first hyperpoljirizability) and to 7 (i.e. 7, the second hyperpolarizability) the problem of cavity errors become worse and worse. 

In the following the polarizability and the first and second hyperpolarizabilities for urea calculated at the SCF level in vacuo and in water are reported. Both static and frequency dependent nonlinear properties have been calculated, with the Coupled Perturbed Hartree-Fock (CPHF) and Time Dependent-CPHF procedures that have been described above. The solvent model is the Polarizable Continuum Model (PCM) whereas vibrational averaging of the optical properties along the C-0 stretching coordinate has been obtained by the DiNa package both in vacuo and in solution. 

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