We have calculated the second hyperpolarizability gamma of neon using the CCSD(T) method. The accuracy of the CCSD(T) approach has been established by explicit comparison with the single, double and triple excitation coupled-cluster (CCSDT) method using extended basis sets that are known to be adequate for the description of gamma. Our best estimate for gamma(sub 0) of 110 +/- 3 a.u. is in good agreement with other recent theoretical values and with Shelton's recent experimental estimate of 108 +/- 2 a.u. Comparison of the MP2 and CCSD(T) hyperpolarizability values indicates that MP2 gives a very good description of the electron correlation contribution to gamma(sub 0). We have combined MP2 frequency-dependent corrections with the CCSD(T) gamma(sub 0) to yield values of gamma(-2 omega;omega,omega,0) and gamma(exp K)(-omega;omega,0,0)

Almloef, Jan

Lee, Timothy J.

Rice, Julia E.

Scuseria, Gustavo E.

Taylor, Peter R.

English

NASA Technical Reports Server

Connected Triple Excitations in Coupled-Cluster
Calculations of Hyperpolarizabilities: Neon
N 9 3 - 1 1_85 :_
Julia E. Rice
IBM Research Division, Almaden Research Center
650 Harry Rd., San Jose, CA 95120-6099
Gustavo E. Scuseria
Department of Chemistry
Rice University, Houston, TX 77251-1892
Timothy J. Lee
NASA Ames Research Center
Moffett Field, CA 94035
Peter R. Taylor t
ELORET Institute
O
Palo Alto CA 94303
and
Jan AlmlSf
Department of Chemistry and Supercomputer Institute
University of Minnesota, Minneapolis, MN 55455
Mailing address: NASA Ames Research Center, Moffett Field, California 94035-1000
https://ntrs.nasa.gov/search.jsp?R=19930002666 2020-03-17T09:50:54+00:00Z
Abstract
We have calculated the second hyperpolarizability 7 of neon using the CCSD(T)
method. The accuracy of the CCSD(T) approach has been established by explicit compar-
ison with the single, double and triple excitation coupled-cluster (CCSDT) method using
extended basis sets that are known to be adequate for the description of 7. Our best es-
timate for 7o of 110-¢-3 a.u. is in good agreement with other recent theoretical values and
with Shelton's recent experimental estimate of 108q-2 a.u. Comparison of the MP2 and
CCSD(T) hyperpolarizability values indicates that MP2 gives a very good description of
the electron correlation contribution to 7o. We have combined MP2 frequency-dependent
corrections with the CCSD(T) 70 to yield values of 7(-2w;w,w, O) and 7K(--w;w, O, 0).
1. Introduction
Theoretical determination of hyperpolarizabilities has been a topic of much interest
recently, since knowledge of atomic and molecular hyperpolarizabilities is central to the
understanding of the non-linear response of matter to light. In particular, organic materials
with large hyperpolarizabilities are candidates for applications such as optical switching
and second harmonic generation, and there is great potential for interaction between theory
and experiment in the study of these systems.
From a theoretical point of view, it is important to understand the requirements for
determining accurate hyperpolarizabilities for small systems, because it is possible to use
large one-particle basis sets and sophisticated electron correlation treatments for these
species, and thereby to evaluate the effects of approximations that will be necessary for
the study of the hyperpolarizabilities of larger systems. Hence for small systems it is
desirable to estimate the accuracy of the calculated hyperpolarizability. This may be
accomplished in two ways. First, the quality of the one and n-particle approximations
used in the calculation can be systematically improved and the convergence of the result
can be monitored. This is perhaps the preferred approach from a theoretical standpoint.
Alternatively, the theoretical value can be compared directly with experiment, although
the possibility of error cancellation between the one and n-particle approximations must
always be borne in mind.
Study of the hyperpolarizabilities of the rare gas atoms has a number of advantages.
In particular, for neon sophisticated levels of theory and large one-particle basis sets can
be employed. Experimental gas-phase electric-field-induced second harmonic generation
data for the rare gases is available over a range of frequencies 1, which makes extrapolation
to the static limit possible for the purposes of comparison with a theoretical static value.
In addition, vibrational effects (which have been shown to be non-negligible for some
molecular values, see, for example, Refs. 2 and 3) vanish for atoms.
3
It must be noted, however,that it is a non-trivial task to demonstrate convergence
of calculatedvaluesfor hyperpolarizabilities. Previous work has shownthat there can be
a much larger electron correlation contribution to hyperpolarizabilities than for linear po-
larizabilities, and that the contribution of higher excitations is not insignificant (see,for
example,Refs. 4-6). For example, in the caseof 7 of neon, where the effect of electron
correlation is about 40 a.u., or 40% of 7, the perturbational estimate of triple excita-
tions contributes 8 a.u. or 20% of the total electron correlation contribution. 4 Since the
contribution of higher excitations is so large, the applicability of approximate methods
for estimating the effects of higher excitations in hyperpolarizability calculations might
be questioned. For example, the single and double excitation coupled-cluster method in-
cluding an estimate of triple excitations through the fourth and partially the fifth order
of perturbation theory, (CCSD(T)) 7 has had great success in describing the structure and
frequencies of a number of 'difficult' chemical systems -- that is, systems whose wave
functions are not dominated by a single reference. 8'9 This success notwithstanding, it is
essential to investigate the utility of this approach specifically for determining accurate
hyperpolarizabilities.
The reliability of a correlation treatment is best evaluated by comparison with a full
configuration-interaction (CI) calculation in the same one-particle basis set. However, since
even at the self-consistent field (SCF) level of theory diffuse f type functions contribute
10 a.u. to the hyperpolarizability of Ne 4, a full CI calibration in a realistic basis set is not
feasible. Here, in order to establish the accuracy of the CCSD(T) method for 7 of neon,
we compare instead to results obtained with the full single, double and triple excitation
coupled-cluster method (CCSDT), x°'1_ in a basis set which is known to be adequate for
the description of hyperpolarizabilities. We can thus assess the accuracy of the computed
CCSD(T) value, as well as compare it with experimental and other theoretical values. We
note that a previous study has demonstrated that for correlation energies, CCSD(T) is an
excellent approximation to CCSDT. 12
2. Computational Methods
The one-particle basis sets used in this work are similar to those employed previously. 4
They were derived from van Duijneveldt's (13s8p) primitive set _z augmented with a (6d4f)
polarization set, with exponents chosen as an even-tempered sequence a= 2.5'_a0; n
= O,...,k with a0 = 0.20, 0.61 for the d and f functions, respectively. This was con-
tracted to [4s 3p 2d If ] using atomic natural orbitalsJ 4 The two outermost sets of
spd functions and the outermost f function were uncontracted to give basis C, denoted
[4+1+1s 3+l+lp 2+1+1d l+lf ]. We use the notation C in order to be consistent with
our previous study. 4 Additional diffuse functions were then added by extrapolating from
the outermost function in an even-tempered sequence, a= 2.5-'_a0. For example, the ad-
dition of one set of diffuse functions is denoted + (lslpldlf). In some calculations the
basis was further augmented with two diffuse 9 functions with exponents of 0.29 and 0.11.
Basis set C was completely uncontracted, and two tight d functions were included (ad
= 123.53, 49.41) for calculations in which core correlation was included. Only the true
spherical harmonic components of the basis functions were used throughout.
Energies were calculated using self-consistent field (SCF), single and double excitation
coupled-cluster (CCSD) and second-order M¢ller-Plesset perturbation theory (MP2). The
effect of triple excitations was investigated using (a) the CCSD(T) method, which includes
an estimate of the triples through fourth and partially fifth order of perturbation theory,
based on the CCSD amplitudes in the perturbation energy expressions; (b) the CCSDT
method, which exphcitly includes all single, double and triple excitations; and (c) the
CCSD + T(CCSD) method as which includes only the fourth-order perturbation theory
contribution based on CCSD amplitudes.
The dipole polarizabilities are defined 16 by the energy response to an applied electric
5
field of strength F:
1 1 1
E(F) = E0- 2aF 2 - _.rTF 4 - _.eF 6
Electric fields of 0, 0.002, 0.004, 0.008, 0.012, 0.016 and 0.020 a.u. were applied and
the energy responses were fitted to the sixth-order polynomial in the field strength. The
SCF value of 7 obtained from the fit agrees with the SCF value obtained from finite
displacements of analytic/3 values to within 0.1 a.u. Tests of the fit for the correlated
values indicate the error in 7 due to the fit is less than 0.2 a.u.
The SCF, MP2 and most of the coupled-cluster calculations were performed using
the MOLECULE-SWEDEN, 1_ CADPAC, is, and TITAN 19 programs. The CCSDT calcu-
lations were performed with a program written by one of us (GES). 11 The SCF energies
were converged to 10 -13 Eh or better and the CCSD and CCSDT energies to 10 -12.
3. Results and Discussion
The values for the linear polarizability, a and the hyperpolarizability, 7 determined
in this work are summarized in Table 1. We note first that the SCF, MP2, CCSD and
CCSD(T) results for both a and 7 are essentially identical in basis sets C+(3s3p2d3f) and
C+(2s2pld2f). This establishes that these values are converged with respect to further
addition of diffuse s, p, d,and f functions in the one-particle basis set. It also verifies
that basis C+(2s2pld2f) is a good choice for comparison of the CCSDT and CCSD(T)
results. The CCSDT 7 value of 110.9 a.u. in this basis is in very good agreement with
that from the CCSD(T) method (111.2 a.u.). This establishes that the (T) correction is
a reliable estimate of the contribution from connected triple excitations to the hyperpo-
larizability of Ne, even though the contribution is large. Comparison of the CCSD(T)
and CCSD + T(CCSD) results with the C+(3s3p2d3f) basis shows that the fourth-order
correction alone is a substantial overestimate.
The effects of diffuse higher-order angular momentum functions and of core correla-
tion have also been investigated at the CCSD(T) level of theory, in order to improve upon
6
our previous best computed value of 111 a.u.4 and to assessfurther our error estimate of
+4 a.u. Uncontraction of the one-particle basis set reduces 7 from 111.0 a.u. to 110.3 a.u.,
and core correlation gives a further reduction of 0.5 a.u. Conversely, diffuse g functions
increase 7 by 0.2 a.u. These changes are very similar to the values of -0.7, -0.5, and
0.2 a.u., respectively, obtained at the CCSD level of theory, whereas the MP2 values indi-
cate that second-order perturbation theory may underestimate the effects of uncontraction
(-0.4 a.u.) and core correlation (-0.3 a.u.). This is consistent with the results for argon, 2°
where the CCSD estimate of the reduction in 7 due to core correlation is larger than the
MP2 value. The CCSD(T) results thus support our earlier conclusions 4 that the combina-
tion of further augmentation of the one-particle basis set and further improvement in the
treatment of core correlation is likely to have little effect on 7.
The importance of higher than triple excitations also requires discussion. Single and
double excitations increase 7 by 34 a.u. and triple excitations by a further 9 a.u. However,
the coupled-cluster expansion should be rapidly convergent, since all the corresponding
disconnected terms are included to infinite order at each level of CC theory. The next
most important contribution comes from connected quadruple excitations: these can be
expected to contribute substantially less than connected triple excitations. It is highly
unlikely that these terms contribute as much as 3 a.u. We have previously established 4
that the CCSD(T) energy, at least, agrees almost perfectly with full CI benchmark results
for Ne. We thus arrive at a best estimate of 110 a.u., with an uncertainty of 3 a.u. The
latter is almost entirely due to uncertainty in the estimated contribution of connected
quadruple (and higher) excitations. We can compare our result with Shelton's most recent
estimate of 108+2 a.u. 2a based on extrapolation of electric-field-induced second harmonic
generation values. This differs from his earlier static value of 119d=2 due to his discovery
of a systematic error in some of the measurements. 21 Our value is also in good agreement
with other recent theoretical values. These include Maroulis and Thakkar's estimate of
114+9 a.u. based on the double-excitation coupled-cluster method augmented with a
perturbational estimate of single and triple excitations, denoted CCD+ST(CCD), 22 as
well as Chong and Langhoff's CCSD(T) value of 111.0 a.u. 23, and 1084-5 a.u. from the
restricted active space self-consistent field calculations of Jensen and co-workers. 24
It is dearly much easier to demonstrate convergence for a than for 7. The electron
correlation contribution is smaller (around 11%) for a and the CCSD(T) and CCSDT
values agree to 3 decimal places. The diffuse function requirement is also less stringent
than for 7. The largest remaining corrections are for core correlation and uncontraction
of the one-particle basis set which decrease a by 0.013 a.u., or 0.5%. Our best computed
value for a is 2.677 a.u. to which we conservatively assign an uncertainty of 0.015 a.u.
This is in excellent agreement with the value of 2.669 a.u. derived from dipole oscillator
strength distributions by Kumar and Meath. 25
Finally, the rather good agreement between the MP2 and CC polarizability and hy-
perpolarizability suggests that the former method may be useful even where, as is the case
for neon, the correlation contributions are large. Because it is much simpler to evaluate
dynamic polarizabilities with MP2 than with CC methods, this suggests that an efficient
route to reliable frequency-dependent results may be to correct CC static values with MP2
differences between static and dynamic values. It is certainly the case that the MP2 values
in themselves would be much more refiable than values obtained using SCF methods to
describe the frequency dependence. We may also note that the experimental frequencies
are far from resonance, so an error in the frequency-dependent hyperpolarizability arising
from the error in the prediction of the poles at the MP2 level of theory is not a matter for
concern.
In this work we have combined the MP2 frequency-dependent corrections 2_ for
7(-2w; w, w, O) of neon with our best estimate for the static value and arrive at
1124-3 a.u.(A=1319 nm), 1134-3 a.u.(A=1064 rim), 124±4 a.u.(A=514.5 nm), to
be compared with experimental second harmonic generation values of 111.1+0.8 a.ufl,
109.9±0.5 a.u. 1 and 122.2±0.5 a.u. 21, respectively. The theoretical results show no nega-
tive dispersion (note that the uncertainties in the theoretical numbers are not independent
and are likely to be strongly correlated with one another), and are also in good agreement
with the revised experimental values. We may also combine the MP2 frequency-dependent
correction 26 for 7K(--w;_,0,0) of neon with our best estimate of the static value and ob-
tain 113±3 a.u. for A =632.8 nm. This value is somewhat higher than the Kerr effect
measurement of 1014-8 a.u. 27
4. Conclusions
We have demonstrated by comparison with full CCSDT results that the CCSD(T)
method provides an accurate description of the hyperpolarizability of neon. Since the
CCSD(T) method has the advantage of being rather inexpensive, it allows extensive inves-
tigation of the one-particle basis set requirements and the core correlation treatment for
cz and 7 of neon. Our best computed CCSD(T) value for 7 of neon, including the effects
of multiple sets of diffuse s, p, d and f functions and of core correlation, is 109.8 a.u.
After incorporating corrections for diffuse g functions and a more complete treatment of
triple excitations our best estimate is 110 a.u. with an error estimate of 3 a.u. This re-
sult is in agreement with all of the most recent theoretical values 22-24 and in line with
the static value extrapolated from experimental frequency-dependent measurements. TM
Together with our assignment of uncertainty, this fulfils our criteria for establishing the
accuracy of the computed value.
References
i
1. D.P. Shelton, Phys. Rev. A 42, 2578 (1990).
2. D.M. Bishop, Rev. Mod. Phys. 62,343 (1990).
3. D.M. Bishop and B. Kirtman, J. Chem. Phys. 95, 2646 (1991).
4. P.R. Taylor, T.J. Lee, J.E. Rice and J. AlmlSf, Chem. Phys. Left. 163, 359 (1989);
Chem. Phys. Left. xxx, xxx (1991).
5. R.J. Bartlett and G.D. Purvis, Phys. Rev. A20, 1313 (1979).
6. G. Maroulis and A.J. Thakkar, J. Chem. Phys. 93,652 (1990).
7. K. Raghavachari, G.W. Trucks, J.A. Pople and M. Head-Gordon, Chem. Phys. Left.
157,479 (1989).
8. T.J. Lee and G.E. Scuseria, J. Chem. Phys. 93,489 (1990).
9. T.J. Lee, A.P. Readell and P.R. Taylor, J. Chem. Phys. 92,489 (1990).
10. J. Noga and R. J. Bartlett, J. Chem. Phys. 85, 2779 (1986).
11. G. E. Scuseria and H. F. Schaefer, Chem. Phys. Lett. 152,382 (1988).
12. G. E. Scuseria and T. J. Lee, J. Chem. Phys. 93, 5851 (1990).
13. F. B. van Duijneveldt, IBM Publication RJ945 (1971).
14. J. AlmlSf and P.R. Taylor, J. Chem. Phys. 86, 4070 (1987).
15. M. Urban, J. Noga, S.J. Cole and R.J. Bartlett, J. Chem. Phys. 83, 4041 (1985).
16. A.D. Buckingham, Adv. Chem. Phys., 12,107 (1967).
17. MOLECULE-SWEDEN is an electronic structure program system written by J.
AlmlSf, C. W. Bauschlicher, M. R. A. Blomberg, D. P. Chong, A. Heiberg, S. R.
Langhoff, P.-/_. Malmqvist, A. P. Rendell, B. O. Roos, P. E. M. Siegbahn, and P. R.
Taylor.
18. R. D. Amos and J. E. Rice, CADPAC:Cambridge Analytic Derivatives Package, Issue
4.0 (Cambridge University, Cambridge, England, 1987).
10
19. TITAN is a set of electronic structure programs written by T.J. Lee, A.P. Rendell and
J.E. Rice.
20. 3.E. Rice, P.R. Taylor, T.3. Lee and 3. AlmiSf, 3. Chem. Phys. 94, 4972 (1991).
21. D.P. Shelton, personal communication and work presented at the North American
Chemical Congress (ACS), August, 1991.
22. G. Maroulis and A.:I. Thakkar, Chem. Phys. Lett. 156, 87 (1989).
23. D.P. Chong and S.R. Langhoff, 3. Chem. Phys. 93, 570 (1990).
24. H.:I. Aa. 3ensen, P. :iCrgensen, H. ttettema and 3. Olsen, Chem. Phys. Lett., in press.
25. A. Kumar and W.:I. Meath, Can. 3. Chem., 63, 1616 (1985).
26. :I.E. Rice, 3. Chem. Phys., submitted.
27. A. D. Buckingham and D. A. Dunmur, Trans. Faraday Soc., 64, 1776 (1968).
11
Table 1
Neon dipole polarizabilities (a.u.)
Basis set Method a 7
C '_ + (2s2pld2f) SCF 2.377 68.66
MP2 2.713 110.6
CCSD 2.643 102.2
CCSD(T) 2.690 110.9
CCSDT 2.690 111.2
C '_ + (3s3p2d3f) SCF 2.377 68.68
MP2 2.713 110.7
CCSD 2.643 102.2
CCSD + T(CCSD) 2.703 115.3
CCSD(T) 2.690 111.0
C _ + (3s3p2d3f2g) SCF 2.377 68.67
MP2 2.716 110.9
CCSD 2.645 102.4
CCSD(T) 2.692 111.2
(13s8p8d4f)+(3s3p2d3f)b SCF 2.377 68.67
MP2 2.708 110.3
CCSD 2.636 101.5
CCSD(T) 2.684 110.3
(13sSp8d4f)+(3_3p2d3f)c MP2 2.703 110.0
CCSD 2.628 100.8
CCSD(T) 2.677 109.8
a See text for the definition of C.
b Uncontracted basis with 8 electrons correlated
Uncontracted basis with 10 electrons correlated
12


Connected triple excitations in coupled-cluster calculations of hyperpolarizabilities: Neon

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