Response functions cubic

In a recent publication [22] we reported the implementation of dispersion coefficients for first hyperpolarizabiiities based on the coupled cluster quadratic response approach. In the present publication we extend the work of Refs. [22-24] to the analytic calculation of dispersion coefficients for cubic response properties, i.e. second hyperpolarizabiiities. We define the dispersion coefficients by a Taylor expansion of the cubic response function in its frequency arguments. Hence, this approach is... 

In the next section we derive the Taylor expansion of the coupled cluster cubic response function in its frequency arguments and the equations for the required expansions of the cluster amplitude and Lagrangian multiplier responses. For the experimentally important isotropic averages 7, 7i and yx we give explicit expressions for the A and higher-order coefficients in terms of the coefficients of the Taylor series. In Sec. 4 we present an application of the developed approach to the second hyperpolarizability of the methane molecule. We test the convergence of the hyperpolarizabilities with respect to the order of the expansion and investigate the sensitivity of the coefficients to basis sets and correlation treatment. The results are compared with dispersion coefficients derived by least square fits to experimental hyperpolarizability data or to pointwise calculated hyperpolarizabilities of other ab inito studies. 

As a consequence of the time-averaging of the quasienergy Lagrangian, the derivative in the last equation gives only a nonvanishing result if the frequencies of the external fields fulfill the matching condition Wj = 0. In fourth order Eq. (29) gives the cubic response function ... 

An implementation of the cubic response function Eq. (30) for the coupled cluster model hierachy CCS, CC2 and CCSD was reported in Ref. [24]. 

In the normal dispersion region below the first pole, response functions can be expanded in power series in their frequency arguments. The four frequencies, associated with the operator arguments of the cubic response function are related by the matching condition a -fwfl +wc -t-U ) — 0. Thus second hyperpoiarizabiiities or in general cubic response properties are functions of only three independent frequency variables, which may be chosen as u>b, ljc and U > ... 

There have been a few recent studies of the corrections due to nuclear motion to the electronic diagonal polarizability (a ) of LiH. Bishop et al. [92] calculated vibrational and rotational contributions to the polarizability. They found for the ground state (v = 0, the state studied here) that the vibrational contribution is 0.923 a.u. Papadopoulos et al. [88] use the perturbation method to find a corrected value of 28.93 a.u. including a vibrational component of 1.7 a.u. Jonsson et al. [91] used cubic response functions to find a corrected value for of 28.26 a.u., including a vibrational contribution of 1.37 a.u. In all cases, the vibrational contribution is approximately 3% of the total polarizability. 

All the linear and nonlinear optical properties introduced above are therefore expressed in terms of linear, quadratic and cubic response functions. They can be computed with high efficiency using analytical response theory [9] with a variety of electronic structure models [8],... 

In the case of the cubic response function we can, using the spectral representation, write the cubic response function as... 

The cubic response function has poles at frequencies given by... 

Here, we provide the theoretical basis for incorporating the PE potential in quantum mechanical response theory, including the derivation of the contributions to the linear, quadratic, and cubic response functions. The derivations follow closely the formulation of linear and quadratic response theory within DFT by Salek et al. [17] and cubic response within DFT by Jansik et al. [18] Furthermore, the derived equations show some similarities to other response-based environmental methods, for example, the polarizable continuum model [19, 20] (PCM) or the spherical cavity dielectric... 

We will now detail the explicit contributions that are due to the PE potential to the cubic response function, which enter through the E matrix and the V"1 vector. The contributions to the E matrix are defined through... 

B. Jansik, P. Salek, D. Jonsson, O. Vahtras, H. Agren, Cubic response functions in time-dependent density functional theory, J. Chem. Phys. 122 (2005) 054107. 

Adiabatic connection formula, 409 Basis Set Superposition Error (BSSE), 67 Cubic response function, 261 ... 

A correct form of the treatment of the long-range asymptotic behaviour of DFT is essential for the investigation of extended systems and this problem has been addressed by Baer and Neuhauser.30 Jansik et al.203 have extended the treatment time-dependent DFT to deal with cubic response functions and have applied their method to nitrogen, benzene and C-60 fullerene. 

The ab initio calculation of NLO properties has been a topic of research for about three decades. In particular, response theory has been used in combination with a number of electronic structure methods to derive so-called response functions [41 8], The latter describe the response of a molecular system for the specific perturbation operators and associated frequencies that characterize a particular experiment. For example, molecular hyperpolarizabilities can be calculated from the quadratic and cubic response functions using electric dipole operators. From the frequency-dependent response functions one can also determine expressions for various transition properties (e.g. for multi-photon absorption processes) and properties of excited states [42]. 

Examples of NLO processes that involve also cubic response functions are ... 

Figure 4. Static hypermagnetizability anisotropy. Atj(O), computed with the d-aug-cc-pV5Z basis set (Neon) and d-aug-cc-pVQZ basis set (Argon). Orbital-relaxed results obtained with a finite field approach from analytically evaluated magnetizabilities are compared to those obtained from orbital-unrelaxed quadratic and cubic response functions...

In the following, we shall derive and implement expressions for the linear, quadratic, and cubic response functions ((A V >)) , V i, and... 

This equation holds for any time-independent one-electron operator Q. In particular, it holds for the spin-averaged excitation operators Ep in the expansion of K(f) in Eq. (41). Collecting these operators in the column vector q, we arrive at a set of nonlinear equations from which the time-dependence of k(t) may be determined. In the following, we shall use these equations to determine the first- and second-order terms in Eq. (46) and thereby the linear, quadratic, and cubic response functions. 

The cubic response functions has poles where the frequency parameters or then-sum matches an excitation energy. As for the quadratic response function we may also consider double residues where two poles matches at the same time (triple residues turn out not to give anything new). 

The cubic response function is obtained as the fourth derivative of the time-averaged quasi-energy. Thus we expand the energy to fourth order in the first- and second-order parameters. 

Time averaging cancels the time-dependent phase factor if the sum of the frequencies is zero. The exchange-correlation energy contribution to tire cubic response function then becomes ... 

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