Cauchy moment

Cauchy Moments of Ne, Ar, and Kr Atoms Calculated Using the Approximate Coupled Cluster... 

The Cauchy moments are derived and implemented for the approximate triples model CC3 with the proper N scaling (where N denotes the number of basis functions). The Cauchy moments are calculated for the Ne, Ar, and Kr atoms using the hierarchy of the coupled-cluster models CCS, CC2, CCSD, CC3 and a large correlation-consistent basis sets augmented with diffuse functions. A detailed investigation of the one- and A-electron errors shows that the CC3 results have the accuracy comparable to the experimental results. 

The calculation of frequency-dependent linear-response properties may be an expensive task, since first-order response equations have to be solved for each considered frequency [1]. The cost may be reduced by introducing the Cauchy expansion in even powers of the frequency for the linear-response function [2], The expansion coefficients, or Cauchy moments [3], are frequency independent and need to be calculated only once for a given property. The Cauchy expansion is valid only for the frequencies below the first pole of the linear-response function. 

The results presented in this chapter are complementary to the ones of Ref. [4], where the Cauchy moments were calculated for the Ne atom using the CCS, CC2, CCSD hierarchy. A systematic improvement in the quality of the Cauchy... 

In the next section, we recapitulate the derivation of the Cauchy moment expressions for CC wavefunction models and give the CC3-specific formulas we also outline an efficient implementation of the CCS Cauchy moments. Section 3 contains computational details. In Section 4, we report the Cauchy moments calculated for the Ne, Ar, and Kr gases using the CCS, CC2, CCSD, CCS hierarchy and correlation-consistent basis sets augmented with diffuse functions. In particular, we consider the issues of one- and A-electron convergence and compare with the Cauchy moments obtained from the DOSD approach and other experiments. 

The Cauchy moments have been derived in Ref. [4] for CC wavefunctions, using the time-dependent quasi-energy Lagrangian technique [I]. In Section 2.1 we recapitulate the important points of that derivation and use it in Section 2.2 to derive the CC3-specific formulas. 

The oscillator strength sums, or Cauchy moments, for an exact state are... 

Calculation of the Cauchy moments, 5ab( 2 2), thus only requires solving a set of frequency-independent linear equations, equation (18), up to order 2k and carrying out the linear transformations involving rf vector and F matrix (see equation (20)). Note, that contrary to equation (1) no sum over states is involved in this formulation of the Cauchy moments. 

To calculate the Cauchy moments we first need to solve the equations for the Cauchy vectors (equation (18)). In the CC3 model this equation may be written as... 

Since the Cauchy moments formula, equation (20), has the same structure as the CC linear-response function, equation (4), the contractions in equation (30) may be implemented by a straightforward generalization of the computational procedures described in Section III B of Ref. [21] for the calculation of the CC3 linear-response function. 

All the calculations have been carried out using a local version of the Dalton program package [26]. The implementation of the Cauchy moments for the CCS, CC2, and CCSD models has been described in Ref. [4]. The CC3 Cauchy moments have been implemented by us following the outline presented in the previous section. 

In Table 1 we have collected the Cauchy moments, S(k), calculated for the Ne atom using the CCSD model and the n-aug-cc-pVAZ basis-set family. As can be seen from the table, single augmentation is not sufficient for the calculation of the Cauchy moments. On the other hand, going beyond the double augmentation... 

Table 1. The basis-set convergence of the Cauchy moments S(k) [a.u.] for Ne calculated with CCSD model and the n-aug-cc-VXZ basis-set family (all electrons correlated)...

The Cauchy moments of Ne at the CCSD/q-aug-cc-pV5Z level were found in Ref. [4] to be converged within 1 % compared to the basis-set limit result. We have calculated the Cauchy moments also for the X—6 cardinal number. From the results in Table 1 it appears that the Cauchy moments at this level are significantly less than 1 % from the basis-set limit result. 

Table 2. The Cauchy moments S(k) for Ne calculated at various levels of the CC hierarchy using the d-aug-cc-pV6Z basis-set (all electrons correlated). All numbers in a.u.

Turning our attention to the CC3 model we observe that the inclusion of triple excitations further improves the quality of the Cauchy moments in the sense that all the CC3 moments are closer to the DOSD results than the CCSD moments. The CC3 and DOSD Cauchy moments are in fact so close that it is worth investigating in more detail the errors associated with the CC3 results. 

Let us now consider the errors in the CC3 S( — 2) Cauchy moment (the static polarizability). From the monotonic convergence of the CCSD doubly augmented basis-set calculations of S( — 2) in Table 1 it appears that the difference between the 5Z and 6Z results should give a good estimate of the CCSD basis-set error at the 6Z level. CC3/d-aug-pV5Z 5 ( —2) Cauchy moment is 2.670 a.u. Using the... 

The Cauchy moments of Kr (Table 4) have been calculated using the largest currently available correlation-consistent basis augmented with diffuse functions - d-aug-cc-pV5Z basis. As for the Ar atom, the CCS and CC2 results overestimate the DOSD values for smaller k and underestimate for larger k. The CCSD model behaves in the same manner. The CC3 model systematically... 

The high quality of the CC3 model and the convergence in one-electron space have allowed us to determine the Cauchy moments for Ne which have the accuracy comparable to the experimental results. To arrive at the similar conclusion for the Ar and Kr atoms a more detailed investigation of the convergence in one-electron and A-electron space is required. The relativistic contribution to the Cauchy moments may also need to be taken into account for these two atoms. 

The LC scheme was also applied to overestimations of polarizabihties in DFT as shown in Table 20.10 [114]. The Sadlej valence triple zeta basis set was used [116,117]. The table shows that calculated polarizabilities of LC-BOP are obviously more accurate than those of BOP. Compared to B3LYP results, LC-BOP shows more improvements in many cases. Similar results were found in calculations of anisotropies and S 4 Cauchy moments of polarizabilities. Hence, we may say that these overestimations also come from the lack of long-range interactions in exchange functionals. 

A variety of techniques has been employed for the estimation of dispersion coefficients. Good reviews are those by Dalgarno and Davison [31] and Dalgarno [32]. The semi-empirical methods based on oscillator strengths were refined by using sum rules for Cauchy moments. A Cauchy moment S(k) is defined by the following sum over... 

The even Cauchy moments arise in the expansion of a frequency-dependent polarizability a((o)... 

Goebel D, Hohm U (1996) Dipole polarizability, Cauchy moments, and related properties of Hg. 1 Phys Chem 100 7710-7712... 

Mata et al have determined the dynamic polarizability and Cauchy moments of liquid water by using a sequential molecular dynamics(MD)/ quantum mechanical (QM) approach. The MD simulations are based on a polarizable model of liquid water while the QM calculations on the TDDFT and EOM-CCSD methods. For the water molecule alone, the SOS/TDDFT method using the BHandHLYP functional closely reproduces the experimental value of a(ffl), provided a vibrational correction is assumed. Then, when considering one water molecule embedded in 100 water molecules represented by point charges, a(co) decreases by about 4%. This decrease is slightly reduced when the QM part contains 2 water molecules but no further effects are observed when enlarging the QM part to 3 or 4 water molecules. These molecular properties have then been employed to simulate the real and imaginary parts of the dielectric constant of liquid water. 

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