Implementation of the incremental scheme for one-electron first-order properties in coupled-cluster theory

(1)

Implementation of the incremental scheme for one-electron first-order properties in coupled-cluster theory

Joachim Friedrich,^1,a兲Sonia Coriani,^2,3Trygve Helgaker,³and Michael Dolg¹

1Institute for Theoretical Chemistry, University of Cologne, Greinstr. 4, 50939 Cologne, Germany

2Dipartimento di Scienze Chimiche, Università degli Studi di Trieste, via Licio Giorgieri 1, I-34127 Trieste, Italy

3Department of Chemistry, Centre for Theoretical and Computational Chemistry (CTCC), University of Oslo, P.O. Box 1033, Blindern N-0315, Norway

共

Received 15 July 2009; accepted 16 September 2009; published online 15 October 2009

兲

A fully automated parallelized implementation of the incremental scheme for coupled-cluster singles-and-doubles

共CCSD兲

energies has been extended to treat molecular

共unrelaxed兲

first-order one-electron properties such as the electric dipole and quadrupole moments. The convergence and accuracy of the incremental approach for the dipole and quadrupole moments have been studied for a variety of chemically interesting systems. It is found that the electric dipole moment can be obtained to within 5% and 0.5% accuracy with respect to the exact CCSD value at the third and fourth orders of the expansion, respectively. Furthermore, we find that the incremental expansion of the quadrupole moment converges to the exact result with increasing order of the expansion: the convergence of nonaromatic compounds is fast with errors less than 16 mau and less than 1 mau at third and fourth orders, respectively

共1 mau= 10

⁻³ea₀²

兲; the aromatic compounds converge slowly

with maximum absolute deviations of 174 and 72 mau at third and fourth orders, respectively.

©2009 American Institute of Physics.

关doi:10.1063/1.3243864兴

I. INTRODUCTION

The coupled-cluster

共

CC

兲

approach is capable of provid- ing highly accurate energies and properties of molecular systems^1–11and is therefore frequently used for benchmark- ing more approximate methods such as those based on density-functional theory. However, given the steep scaling of the cost of the standard CC methods with respect to the size of the one-particle basis set, the application of CC theory is usually limited to relatively small molecules. For larger molecules, it becomes necessary to introduce further approximations into the CC model.^12–27 One of these approximations is the incremental scheme of Stoll,^13,28,29which is based on the Bethe–Goldstone expansion introduced into quantum chemistry by Nesbet^12,30,31more than 40 years ago.

During the past 15 years, the incremental scheme has been successfully applied to various closed-shell periodic^32–36and molecular^37–41 systems; extensions to open-shell systems have also been reported.^42–44

Recently, we presented a fully automated implementation of the incremental scheme for molecular electronic energies within the framework of second-order Møller–Plesset perturbation theory

共MP2兲, coupled-cluster singles-and-

doubles

共CCSD兲

theory, and CCSD with a perturbative triples correction.38,39,45,46

Later, we introduced distance screening and an efficient treatment of core-valence correlation.^39,47Given that CC theory provides accurate molecular properties,1,3–5,7–11,48–55

the local domain approxima- tion of Pulay and co-workers14,15,56,57

was used to reduce the cost of such calculations for molecular dipole moments and

for static dipole polarisabilities,⁵⁸ as well as for optical- rotation and magnetic-field parameters.⁵⁹ Within the framework of the Bethe–Goldstone theory, Nesbet³¹ recognized already in 1969 how properties can be treated. Recently, Yang and Dolg⁶⁰ applied this approach to optical tensors of the Ga₄As₄H₁₈cluster.

In the present paper, we examine the accuracy of the incremental scheme for first-order molecular properties such as the dipole and quadrupole moments, which are important quantities characterizing the molecular charge distribution.

II. THEORY

A. The incremental scheme

In an incremental calculation, we divide the total system into domains consisting of groups of localized occupied orbitals and then calculate the correlation energy separately for each domain. Next, to account for the nonadditivity of the domain energies, we calculate correction energies for all pairs of domains, all triples of domains, and so on, until the desired accuracy is reached. The total correlation energy is then computed according to

E_corr=

兺

_i ^⌬␧ⁱ⁺_2!¹

兺

_ij ^⌬␧^ij⁺_3!¹

兺

_ijk^⌬␧^ijk⁺ ^¯^,

⌬␧i=␧i, ⌬␧ij=␧ij−⌬␧i−⌬␧j,

共1兲

where␧iis the correlation energy of the subsystemiand␧ij

is the correlation energy of subsystemsi andjtogether. In a compact set-theory notation, Eq.

共1兲

reads

a兲Electronic mail: [email protected].

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E_corr=

兺

_X

X苸P共D兲∧兩X兩ⱕO

⌬␧_X,

共2兲

where D is the set of domains, P共D兲 is the power set of domains, and Ois the order of the expansion. The summa- tion indexXin Eq.

共2兲

runs over all increments up to order O. When ␧_X represents the correlation energy of the unified subsystems of X, the general correlation-energy increment

⌬␧_Xis given as

⌬␧_X=␧_X−

兺

Y Y苸P共X兲∧兩Y兩⬍兩X兩

⌬␧_Y.

共3兲

For tensorial properties such as the dipole and quadrupole moments, we must express each tensor component in its own incremental expansion. The correlation contribution to the ath component of a tensorial propertyP^atherefore reads

P_corr^a =

兺

_X

⌬P_X^a,

共

4

兲

with the general increment

⌬P_X^a=P_X^a−

兺

_Y

Y苸P共X兲∧兩Y兩⬍兩X兩

⌬P_Y^a.

共

5

兲

For example, the incremental expansion of the dipole moment reads

␮corr=

兺

_X

冢

^⌬^⌬^⌬^␮^␮^␮^X^x^X^y^X^z

冣

^.

^共6兲

B. The local molecular-orbital basis

The incremental scheme requires the wave function to be expressed in a set of local molecular orbitals

共LMOs兲. To use

such a basis in combination with the current implementation of the CCSD first-order one-electron properties in DALTON, which assumes canonical molecular orbitals

共MOs兲,

⁴⁸we use a pseudocanonical basis. This means that we diagonalize the Fock matrix in the LMO basis in the subspace of the domain and use the resulting transformation matrix to generate the pseudocanonical MOs from the LMOs—for details, see Refs.

45and46.

C. First-order one-electron CCSD properties

All properties considered in the present paper have been calculated using an orbital-unrelaxed scheme, in which the MOs are not allowed to adjust

共relax兲

in response to the perturbation. Instead, the response of the wave function to the external perturbation is described entirely by the CC amplitudes. At the CCSD level of theory, for example, unrelaxed first-order one-electron properties are given by⁴

FIG. 1. Structure of the prism isomer of共H₂O兲6taken form Ref.72. The geometries of LiC₅H₇, aniline, furan, andp-nitroaniline were optimized at the RI-BP86/SVP level of theory.

FIG. 2. RI-BP86/SVP optimized structure of benzene and octane.

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具

O

典

CCSD=

兺

_pq^O^pq^D^pq^,

^共

⁷

^兲

whereOpqare the one-electron MO integrals for the operator associated with the given property and the CCSD density- matrix elements take the form

D_pq=

再 ^具

^HF

^兩

⁺

^兺

^␮ ^t¯^␮

^具

^␮

^兩

^exp

^共

⁻^T

^兲冎

^E^pq^exp

^共

^T

^兲兩

^HF

^典 ^共

⁸

^兲

T=T₁+T₂=

兺

ai

t_i^aEai+1 2

兺

ai,bj

t_ij^abEaiEbj.

共9兲

Here theE_pqare excitation operators from orbitalpto orbital q, while t_i^a and t_ij^ab are single and double excitation amplitudes, denoting occupied MOs by iandj, virtual MOs bya andb, and unspecified

共

occupied or virtual

兲

MOs bypandq.

In a more general notation, the cluster operator is given by T=

兺

_␮t_␮␶_␮^{, where}^t_␮are the amplitudes and␶_␮are the excitation operators

兩

␮

典

=␶␮

兩HF典. The CCSD amplitudes

t_␮ are obtained iteratively by solving the projected nonlinear CC equations

具

␮

兩exp共−

T兲Hexp共T兲兩HF典= 0,

共10兲

while the Lagrange multipliers t¯_␮, which have been introduced to make the CC energy stationary with respect to

variations in the amplitudes t_␮, are obtained from the solu- tion of the linear response equations

兺

_␮ ^t¯^␮

^具

^␮

^兩exp共−

^T兲关H,^␶^␯

兴exp共T兲兩HF典

= −

具HF兩关H,

␶␯

兴exp共T兲兩HF典. 共11兲

The density matrix in Eq.

共

8

兲

is unrelaxed since the contributions arising from the relaxation of the canonical MOs to the external perturbation are not included—see, for example, Eq.

共

11

兲

in Ref.4. InDALTON, the first-order CCSD properties are formulated in terms of T₁-transformed one-electron operators⁴

具O典

=

兺

pq

O_pq^⌳D_pq^⌳.

共12兲

Here the one-electron integrals are given by^1,48 O_pq^⌳ = exp共−T₁

兲O

pqexp共T1

兲

=

兺

␣␤

O_␣␤⌳_␣p p ⌳_␤q

h ,

共13兲

⌳^p=C共1−t₁^T

兲,

⌳^h=C共1+t₁

兲共14兲

in terms of the atomic-orbital

共AO兲

integralsO_␣␤ and MO coefficients C. The density-matrix elements in Eq.

共12兲

are given by

TABLE I. Convergence of the incremental CCSD and MP2 correlation energies共s_dsp= 3 , t_con= 3兲. Basis set cc-pVDZ.

Order

CCSD MP2

Energy 共a.u.兲

Error 共kcal/mol兲

Relative 共%兲

Energy 共a.u.兲

Relative 共%兲 Aniline

1 ⫺0.617 241 251.12 60.67 ⫺0.510 440 291.45 52.36

2 ⫺1.041 444 ⫺15.08 102.36 ⫺0.959 490 9.67 98.42

3 ⫺1.017 506 ⫺0.05 100.01 ⫺0.974 373 0.33 99.95

4 ⫺1.017 251 0.11 99.98 ⫺0.974 894 0.00 100.00

Exact ⫺1.017 419 ⫺0.974 901

Benzene

1 ⫺0.515 293 204.38 61.27 ⫺0.425 454 236.61 53.01

2 ⫺0.861 556 ⫺12.90 102.44 ⫺0.789 628 8.09 98.39

3 ⫺0.840 740 0.16 99.97 ⫺0.802 165 0.22 99.96

4 ⫺0.840 858 0.09 99.98 ⫺0.802 534 ⫺0.01 100.00

Exact ⫺0.840 998 ⫺0.802 521

Octane

1 ⫺0.926 438 234.93 71.22 ⫺0.754 940 271.30 63.59

2 ⫺1.312 043 ⫺7.04 100.86 ⫺1.180 383 4.33 99.42

3 ⫺1.300 695 0.08 99.99 ⫺1.187 187 0.06 99.99

4 ⫺1.300 839 ⫺0.01 100.00 ⫺1.187 278 0.00 100.00

Exact ⫺1.300 827 ⫺1.187 278

p-nitroaniline

1 ⫺0.932 523 387.30 60.17 ⫺0.780 900 465.77 51.27

2 ⫺1.586 174 ⫺22.88 102.35 ⫺1.497 365 16.18 98.31

3 ⫺1.550 246 ⫺0.33 100.03 ⫺1.522 604 0.34 99.96

4 ⫺1.549 382 0.21 99.98 ⫺1.523 191 ⫺0.03 100.00

Exact ⫺1.549 719 ⫺1.523 149

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D_pq^⌳ =

具HF兩E

pq

兩HF典

+

具HF兩关E

pq,T₂

兴兩HF典

+

兺

_␮ ^t¯^␮

^具

^␮

^兩exp共−

^T²

^兲E

^pq^exp共T²

^兲兩HF典

^,

^共15兲

with the following explicit expressions:⁴⁸

D_ai^⌳=t¯ai, D_ab^⌳=

兺

_ckl^t^kl^cb^t¯^kl^ca^,

^共16兲

D_ij^⌳= 2␦ij−

兺

cdk

t_ki^cdt¯_kj^cd, D_ia^⌳=

兺

bj

t¯^b_j

共2t

ij

ab−t_ij^ba

兲. 共17兲

For efficiency, the first-order properties in Eq.

共12兲

are calculated as

具O典

=

兺

_␣␤^O^␣␤^D^␣␤^⌳^, ^D^␣␤^⌳ ⁼

兺

_pq ^⌳^p^p^␣^⌳^q^h^␤^D^pq^⌳

^共18兲

following a backtransformation of the density matrix to the AO basis.

III. COMPUTATIONAL DETAILS A. Incremental calculations

In our implementation, the one-electron properties are calculated in the following manner. First, we solve the Hartree–Fock equations for the full system with DALTON

共Ref.

61兲and extract the canonical MOs and the dipole integrals in the AO basis. Next, we localize the orbitals with the

Boys localization,⁶² using a sequence of orthogonal 2⫻2 rotations as introduced by Edmiston and Ruedenberg.⁶³The resulting localized occupied orbitals are grouped into disjoint sets of occupied orbitals as described in Ref.38. This computational step is controlled by two input parameters: the domain-size parameters_dsp, which is a rough measure for the number of LMOs belonging to each of the domains, andt_con, which is a distance parameter for the connectivity of a pair of occupied orbitals.³⁸ The incremental calculations are finally performed with a version of theDALTONprogram package⁶¹ modified to compute automatically the correlation contribution to the given property. In all calculations, all electrons were correlated.

B. Geometries

Unless otherwise stated, the geometries were optimized with the RI-BP86/SVP gradient-corrected Kohn–Sham method,^64,65 as implemented in the TURBOMOLE 5.10

共Refs.

66–69兲 quantum-chemistry package. All stationary points were characterized by analyzing the Hessian matrix.⁷⁰ Note that the goal of the current work is not to derive accurate structural data for the compounds investigated, but rather to obtain reasonable geometrical parameters for the subsequent incremental calculations of energies and properties.

TABLE II. Convergence of the incremental CCSD and MP2 correlation energies共s_dsp= 3 , t_con= 3兲.

Order

CCSD MP2

Energy 共a.u.兲

Relative 共%兲

Energy 共a.u.兲

Relative 共%兲 Furan, aug-cc-pVDZ

1 ⫺0.478 388 188.50 61.43 ⫺0.398 073 224.28 52.69

2 ⫺0.792 792 ⫺8.79 101.80 ⫺0.743 029 7.82 98.35

3 ⫺0.779 215 ⫺0.27 100.06 ⫺0.755 068 0.26 99.94

4 ⫺0.778 794 0.00 100.00 ⫺0.755 469 0.01 100.00

Exact ⫺0.778 786 ⫺0.755 485

Furan, cc-pVDZ

1 ⫺0.466 064 178.42 62.11 ⫺0.386 055 212.16 53.31

2 ⫺0.763 408 ⫺8.16 101.73 ⫺0.712 647 7.22 98.41

3 ⫺0.751 118 ⫺0.45 100.10 ⫺0.723 757 0.25 99.94

4 ⫺0.750 380 0.01 100.00 ⫺0.724 144 0.01 100.00

Exact ⫺0.750 398 ⫺0.724 158

LiC₅H₇, 6-31G^ⴱⴱ

1 ⫺0.560 360 120.29 74.51 ⫺0.480 111 141.56 68.03

2 ⫺0.755 812 ⫺2.36 100.50 ⫺0.702 470 2.03 99.54

3 ⫺0.752 052 0.00 100.00 ⫺0.705 641 0.04 99.99

4 ⫺0.752 052 0.00 100.00 ⫺0.705 699 0.00 100.00

Exact ⫺0.752 056 ⫺0.705 700

共H₂O兲6, aug-cc-pVDZ

1 ⫺1.360 829 15.72 98.19 ⫺1.315 832 16.97 97.99

2 ⫺1.386 000 ⫺0.08 100.01 ⫺1.342 853 0.01 100.00

3 ⫺1.385 865 0.01 100.00 ⫺1.342 869 0.00 100.00

4 ⫺1.385 875 0.00 100.00 ⫺1.342 869 0.00 100.00

Exact ⫺1.385 875 ⫺1.342 869

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C. Hardware

The calculations were performed on a cluster of Intel Core2Quad Q6600 PCs with 2.4 GHz processors, 8 Gbytes random-access memory, and 160 Gbytes disk space per node.

The PCs are connected with a 1 Gbit ethernet.

IV. APPLICATIONS

To obtain a sufficiently broad set of data, we considered a set of chemically relevant systems of different natures, such as a water cluster, an organolithium compound, various aromatic systems

共

aniline, furan, p-nitroaniline, and ben-

TABLE III. Convergence of the incremental dipole moments共in a.u.兲for the dipolar molecules in Fig.1. The parameterst_con= 3 ands_dsp= 3 were used except for共H₂O兲6, wheres_dsp= 5 was used to get a water molecule as a fragment. The cc-pVDZ basis was used except for LiC₅H₇and共H₂O兲6, where the 6-31G^ⴱⴱ basis and the aug-cc-pVDZ basis sets were used, respectively. The RMS deviation of the three dipole components and the MA deviation are used as a measure for the convergence.

Order 兩␮ជtot兩兩␮ជtot兩% 兩␮ជcorr兩 RMS MA

共H₂O兲6, 6 domains 1 0.9982 100.1 0.0519 0.0014 0.0007

2 0.9965 99.9 0.0536 0.0004 0.0005

3 0.9972 100.0 0.0529 0.0000 0.0000

4 0.9972 100.0 0.0530 0.0000 0.0000

Exact 0.9972 100.0 0.0530

LiC₅H₇, 6 domains 1 2.2601 102.2 0.0219 0.0275 0.0467

2 2.2419 101.3 0.0041 0.0170 0.0277

3 2.2148 100.1 0.0265 0.0015 0.0025

4 2.2127 100.0 0.0283 0.0001 0.0001

Exact 2.2126 100.0 0.0284

Aniline, 8 domains 1 0.6247 90.5 0.0724 0.0645 0.0877

2 0.7300 105.7 0.1227 0.0329 0.0484

3 0.7194 104.2 0.1057 0.0244 0.0327

4 0.6869 99.5 0.0667 0.0052 0.0080

Exact 0.6903 100.0 0.0704

Furan, 6 domains 1 0.2047 107.9 0.0890 0.0163 0.0244

2 0.1946 102.6 0.1025 0.0193 0.0333

3 0.1818 95.9 0.1069 0.0046 0.0078

4 0.1905 100.4 0.0982 0.0007 0.0009

Exact 0.1897 100.0 0.0990

p-nitroaniline, 11 domains 1 2.7887 106.4 0.2269 0.0968 0.1673

2 2.6090 99.5 0.4113 0.0345 0.0583

3 2.6135 99.7 0.4021 0.0052 0.0077

4 2.6309 100.4 0.3847 0.0055 0.0096

Exact 2.6214 100.0 0.3942

TABLE IV. Convergence of the incremental dipole moments共in a.u.兲for the nondipolar molecules in Fig.2 共t_con= 3 , s_dsp= 3兲. The RMS deviation of the three dipole components and the MA deviation are used as a measure for the convergence. Basis set cc-pVDZ.

Order 兩␮ជtot兩兩␮ជcorr兩 RMS MA

Benzene, 7 domains 1 0.0429 0.0428 0.0247 0.0353

2 0.0537 0.0537 0.0310 0.0474

3 0.0131 0.0130 0.0075 0.0130

4 0.0033 0.0032 0.0019 0.0031

Exact 0.0000 0.0000

Octane, 10 domains 1 0.0125 0.0125 0.0072 0.0120

2 0.0052 0.0052 0.0030 0.0047

3 0.0012 0.0012 0.0007 0.0010

4 0.0003 0.0003 0.0001 0.0002

Exact 0.0000 0.0000

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zene兲, and octane. The molecules with a permanent dipole moment are given in Fig. 1 and the molecules without a permanent dipole moment are given in Fig.2. Note that for a meaningful study, the compounds must not be too small since the incremental scheme would then yield the exact result by construction—that is, the highest-order increment would require the full calculation.⁷¹

A. Energies

TablesIandIIshow the convergence of the incremental CCSD and MP2 energies for the molecules in Figs.1and2.

In agreement with earlier findings for energies, the convergence of the incremental series is fast. For all systems, the third-order expansion is sufficient to obtain chemical accuracy

共with respect to the exact CCSD and MP2 values兲, with

an error less than 1 kcal/mol in the total energy. Conversely, the second-order expansion does not provide chemical accuracy except for the rapidly convergent water cluster. As reported elsewhere,⁴⁶ the inclusion of diffuse basis functions does not significantly change the energy convergence—see the cc-pVDZ and aug-cc-pVDZ results for furan in TableII.

We note that whereas the first-order MP2 error is always larger than the corresponding CCSD error, the second-order MP2 error is always smaller than the corresponding CCSD error.

The MP2 correlation energies were obtained from the canonical energy expression. Therefore, the convergence of the incremental MP2 energy is a measure of the performance of the pseudocanonical basis. Noting that the MP2 correlation energy converges rapidly for all molecules in this study, we conclude that the pseudocanonical orbitals can be safely used in combination with an implementation based on canonical orbitals.

B. Dipole moments

In Tables III and IV, we compare the incremental and exact CCSD dipole moments for the polar and nonpolar molecules, respectively. The tables contain the magnitude of the total dipole moment at a given order in the incremental expansion

共兩

␮ជtot

兩兲, the magnitude of the total dipole

moment relative to the CCSD exact value

关兩

␮ជtot

兩

%= 100⫻

共兩

␮ជtot

兩

/

兩

␮ជexact

兩兲, only for the polar molecules兴,

and the magnitude of the correlation contribution to the dipole moment

共兩

␮ជcorr

兩兲. As error measures, we have included

the root-mean-square

共

RMS

兲

deviation of the three dipole components and the maximum absolute

共

MA

兲

deviation from the exact CCSD values.

Considering the incremental second-order dipole moments in TablesIIIandIV, the largest errors are observed for the aromatic systems, with RMS deviations of 35 and 33 mau

共1 mau= 10

⁻³ea₀

兲

forp-nitroaniline and aniline, respectively. By contrast, octane and the water cluster converge rapidly, with second-order RMS deviations of only 3 and 0.4 mau, respectively. At third order, all molecules have RMS deviations of around or less than 5 mau, except aniline, which has a large remaining deviation of 24 mau. Finally, at fourth order, the largest RMS deviations are 5–6 mau for aniline and p-nitroaniline, while the remaining molecules have deviations of 2 mau or less.

To investigate the importance of diffuse functions and, in particular, the performance of the incremental scheme for such functions, we have calculated the dipole moment of furan in the larger aug-cc-pVDZ basis—see TableV. Com- paring TablesIIIandV, we find a change in the dipole moment from 190 to 245 mau when diffuse functions are added.

This change is more than one order of magnitude larger than

TABLE V. Convergence of the incremental dipole moments共in a.u.兲for furan using the aug-cc-pVDZ basis 共Fig.1兲. The RMS deviation of the three dipole components and the MA deviation are used as a measure for the convergence.

Order 兩␮ជtot兩兩␮ជtot兩% 兩␮ជcorr兩 RMS MA

s_dsp= 3, 6 domains

1 0.2955 120.7 0.0501 0.0387 0.0476

2 0.2275 93.0 0.0992 0.0324 0.0510

3 0.2408 98.4 0.0659 0.0024 0.0039

4 0.2453 100.2 0.0614 0.0004 0.0005

s_dsp= 4, 4 domains

1 0.2587 105.7 0.0658 0.0245 0.0411

2 0.2439 99.6 0.0845 0.0291 0.0501

3 0.2408 98.4 0.0659 0.0023 0.0040

4 0.2448 100.0 0.0620 0.0000 0.0000

Exact 0.2448 100.0 0.0620

0 10 20 30 40 50 60

2 3 4

abs.dev.ofµiinmau

Order

FIG. 3. Absolute deviations of the individual Cartesian dipole-moment components from the corresponding exact CCSD values,兩␮iCCSD−␮iinc.兩, with respect to the order of the incremental expansion for all molecules in Figs.1 and2.

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the third-order errors in the incremental expansion, which are 8.0 and 4.1 mau for the cc-pVDZ and the aug-cc-pVDZ basis sets, respectively. At fourth order, the dipole moment is converged to an error less than 0.5% in both basis sets. For furan, therefore, the convergence of the incremental dipole- moment expansion is less sensitive to the basis set than is the value of the dipole moment itself.

Concerning the convergence of

兩

␮ជtot

兩

for furan withs_dsp

= 4, we note that the norm of the dipole moment is accurately reproduced at the second order of the expansion, with an error of only 0.4%, compared to an error of 1.6% at third order. The small second-order error is fortuitous and reflects an error cancellation between the different components of the dipole moment, noting that the correlation contribution to the dipole moment converges smoothly and monotonically to- ward the exact result, if the RMS and the MA deviation are used as measure.

Figure3 contains the absolute deviations of all dipole- moment components in this study and illustrates the overall improvement in the description of the dipole moment with increasing order of the incremental expansion. Figures4and 5present the convergence of the incremental dipole moments for the same molecules using MA and RMS deviations, respectively. In both figures, the aromatic systems

共

benzene, aniline, and p-nitroaniline

兲

have the largest errors at third and fourth orders. As expected, the smallest error is observed for the water cluster, with the water molecules as domains

共

s_dsp= 5

兲

.

Comparing our results with those of Koronaet al.,⁵⁸we find that an incremental third-order expansion yields similar

maximum relative errors of about 5%. Note that a fourth- order expansion yields a maximum relative error of only 0.5%. Analyzing the differences in the relaxed to the unrelaxed dipole moments in Ref.11, we find a maximum difference of 27 mau. Comparing this deviation with the RMS deviations of 24 and 6 mau at the third- and fourth-order expansions, respectively, we find that the contribution from orbital relaxation to the dipole moment is typically as large

TABLE VI. RMS and MA deviations for the quadrupole moment共in a.u.兲共t_con= 3 , s_dsp= 3兲. Basis set cc-pVDZ, unless otherwise specified.

Order RMS MA

共H₂O兲6,^a,b6 domains

1 0.0316 0.0571

2 0.0046 0.0088

3 0.0004 0.0004

4 0.0000 0.0000

Aniline, 8 domains

1 0.1504 0.2931

2 0.1206 0.2294

3 0.0135 0.0206

4 0.0086 0.0157

Furan,^b6 domains

1 0.1294 0.2157

2 0.0285 0.0473

3 0.0116 0.0240

4 0.0011 0.0020

Benzene, 7 domains

1 0.1599 0.2860

2 0.1163 0.2208

3 0.0443 0.0743

4 0.0026 0.0039

LiC₅H₇^c, 6 domains

1 0.3061 0.5893

2 0.0585 0.1162

3 0.0058 0.0130

4 0.0004 0.0006

Furan, 6 domains

1 0.0921 0.0818

2 0.0823 0.1547

3 0.0082 0.0140

4 0.0013 0.0026

p-nitroaniline, 11 domains

1 0.3050 0.5967

2 0.4002 0.7275

3 0.0946 0.1743

4 0.0365 0.0724

Octane, 10 domains

1 0.0685 0.1195

2 0.0801 0.1593

3 0.0085 0.0161

4 0.0006 0.0009

as_dsp= 5.

baug-cc-pVDZ.

c6-31G^ⴱⴱbasis.

0 10 20 30 40 50 60

2 3 4

max.abs.dev.ofµiinmau

Order LiC₅H₇ Benzene (H₂O)₆ Octane p-nitroaniline Aniline Furan

FIG. 4. MA deviations of the dipole-moment components of each molecule in Figs.1and2with respect to the order of the incremental expansion.

0 10 20 30 40

2 3 4

RMSofµiinmau

Order LiC₅H₇ Benzene (H₂O)₆ Octane p-nitroaniline Aniline Furan

FIG. 5. RMS deviations of the three dipole-moment components of each molecule in Figs. 1 and2. with respect to the order of the incremental expansion.

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as the third-order contribution in the incremental scheme. We conclude that the incremental scheme can be applied to calculate molecular dipole moments with sufficiently high accuracy.

C. Quadrupole moments

Table VI presents the RMS and MA deviations of the components of the

共traceless兲

quadrupole moment ⌰ij as a measure of the convergence of the incremental series; the full data set is given in the supplementary material.⁷³ The RMS was calculated from six components of the quadrupole tensor. Figures6–8present various plots of the data in Table VI to illustrate the potential accuracy of the incremental scheme for molecular quadrupole moments.

Figure6shows the deviations of the components of the quadrupole moments for the molecules in this study, illustrat- ing the reduced spread of errors with increasing order of the incremental expansion. In Figs. 7 and8, the MA and RMS deviations of the quadrupole-moment components are plot- ted as functions of the expansion order. The slow convergence for p-nitroaniline illustrated in these plots is under- standable since we use a local correlation approach to calculate the electric properties of this push-pull system. As for the dipole moment, the aromatic systems converge slowly and the fastest convergence is observed for the water cluster. Concerning the absolute accuracy, we find that the third-order expansion is sufficient to converge the MA deviation of the quadrupole-moment components to an error less than 16 mau

共

1 mau= 10⁻³ea₀²

兲

for nonaromatic compounds.

For the aromatic compounds, we find third-order MA deviations of 174, 74, and 21 mau for p-nitroaniline, ben- zene, and aniline, respectively; at fourth order, the corresponding errors are reduced to 72, 4, and 16 mau. At fourth order in the expansion, the maximum deviation of the nonaromatic compounds is less than 1 mau. In contrast to the incremental energy expansion, we observe a slower convergence for the quadrupole moment for aromatic compounds compared to the nonaromatic systems.

Comparing the errors introduced by the incremental scheme for furan in cc-pVDZ and aug-cc-pVDZ basis sets, we observe no significant change in the convergence. Com- paring the change in the correlation contribution of⌰11 due to the change in the basis, we find a difference of 196 mau—

much larger than the MA deviation of the third-order incremental values of 14 and 24 mau. In this sense, we can obtain higher accuracy for large systems with the incremental scheme, if the conventional calculation is only possible in a smaller basis set.

Slow convergence can be avoided by increasing the domain sizes. From our experience in incremental energy calculations this improves the speed of the convergence.

V. STATISTICAL ANALYSIS AND COMPUTATIONAL PERFORMANCE

To get an impression of the overall performance of the incremental scheme, we have collected in TableVIIthe RMS and the MA deviations for the entire set of dipole- and quadrupole-moment components of all molecules in this study. Since the quadrupole tensor is symmetric, we have used six symmetry-independent tensor components for each system in the calculation of the RMS deviations.

At third order, the statistical RMS and MA deviation of the dipole-moment components are 10 and 33 mau, respectively; at fourth order, they are converged to 3 and 10 mau, respectively. We conclude that the incremental expansion can be used to calculate dipole moments to a sufficiently high accuracy for molecules similar to those in the present study.

The quadrupole moments are more difficult to converge.

The MA error is completely determined by p-nitroaniline, which shows by far the worst convergence for the quadrupole moment

共vide supra兲. The RMS deviations, which are

also strongly affected by the slow p-nitroaniline conver- gence, are 40 and 14 mau at third and fourth orders, respec-

0 100 200 300 400 500 600 700 800

2 3 4

abs.dev.ofΘijinmau

Order

FIG. 6. Absolute deviation of the components of the QPM兩⌰ijCCSD−⌰ijinc.兩 with respect to the order of the incremental expansion for the molecules in Figs.1and2.

0 100 200 300 400 500 600 700 800

2 3 4

max.abs.dev.ofΘijinmau

Order

LiC₅H₇ Benzene (H₂O)₆ Octane p-nitroaniline Aniline Furan

0 25 50 75 100 125 150

3 4

max.abs.dev.ofΘijinmau

Order

FIG. 7. MA deviation of the quadrupole-moment components with respect to the order of the incremental expansion for the molecules in Figs.1and2. The figure on the right-hand side shows in detail the accuracy at higher orders.

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tively. We conclude that the quadrupole moment can be obtained at reasonable accuracy.

Concerning the performance of the incremental scheme, we note that our current, pilot implementation is not efficient with respect to CPU time. However, the goal of the present work was to examine the convergence of the expansion in Eq.

共4兲

for one-electron first-order properties rather than to attempt the implementation of an efficient production code.

From a technical point of view, Eq.

共4兲

is appealing since it inherently parallelizes the computation with very little com- munication, making it possible to use inexpensive clusters of standard PCs connected by a standard network to evaluate CC energies or properties. If a finite-field approach is used as in Ref.11, the evaluation of the dipole moment reduces to a set of energy calculations, which within the incremental scheme can be carried out very efficiently as discussed in Refs.45and46; in some cases, the incremental scheme was in fact necessary to make the energy calculations at all feasible.⁷⁴ In the present work, we have abandoned the finite-field approach and instead pursued an analytic approach to first-order one-electron properties, opening the route to other first-order properties such as molecular gradi- ents and subsequently to higher-order properties.

VI. CONCLUSION

We have extended the fully automated implementation of the incremental scheme of CC theory to first-order one- electron properties—in particular, molecular dipole and quadrupole moments. We have found that the incremental scheme provides accurate correlation contributions to the dipole and quadrupole moments at low orders of the expansion. The convergence of the incremental expansion of the

dipole and quadrupole moments has been analyzed in terms of the RMS and the absolute maximum deviation of the components. Based on these characteristics, we observe a fast convergence for nonaromatic systems, whereas some aromatic compounds are more difficult to treat. Typically, the errors introduced by a truncation of the expansion at third order are already smaller than the changes observed upon extension of the basis sets from cc-pVDZ to aug-cc-pVDZ quality. Thus, since low-order incremental calculations allow the use of larger basis sets, better overall results are achiev- able with the incremental scheme than with the standard scheme.

ACKNOWLEDGMENTS

This work was supported by a grant from the German Research Foundation

共

DFG

兲

through the priority program 1145 and the SFB 624 and from the Norwegian Research Council through the CoE Center for Theoretical and Compu- tational Chemistry

共CTCC兲共Grant No. 179568/V30兲. Part of

this work was carried out during a visit of J.F. at the CTCC, financed by the CTCC.

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RMSofΘijinmau

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Implementation of the incremental scheme for one-electron first-order properties in coupled-cluster theory