Calculation of electric dipole hypershieldings at the nuclei in the Hellmann–Feynman approximation

Calculation of electric dipole hypershieldings at the nuclei in the Hellmann–Feynman approximation

Alessandro Soncini and Paolo Lazzeretti

Dipartimento di Chimica dell’Universita` degli Studi di Modena, Via Campi 183, 41100 Modena, Italy Vebjørn Bakken and Trygve Helgaker

Department of Chemistry, University of Oslo, P.O. Box 1033, Blindern, N-0315, Oslo, Norway 共Received 12 May 2003; accepted 8 October 2003兲

The third-rank electric hypershieldings at the nuclei of four small molecules have been evaluated at the Hartree–Fock level of theory in the Hellmann–Feynman approximation. The nuclear electric hypershieldings are closely related to molecular vibrational absorption intensities and a generalization of the atomic polar tensors 共expanded in powers of the electric field strength兲 is proposed to rationalize these intensities. It is shown that the sum rules for rototranslational invariance and the constraints imposed by the virial theorem provide useful criteria for basis-set completeness and for near Hartree–Fock quality of nuclear shieldings and hypershieldings evaluated in the Hellmann–Feynman approximation. Twelve basis sets of different size and quality have been employed for the water molecule in an extended numerical test on the practicality of the proposed scheme. The best results are obtained with the R12 and R12⫹basis sets, designed for the calculation of electronic energies by the explicitly correlated R12 method. The R12 basis set is subsequently used to investigate three other molecules, CO, N₂, and NH₃, verifying that the R12 basis consistently performs very well. © 2004 American Institute of Physics.

关DOI: 10.1063/1.1630016兴

I. INTRODUCTION

For a molecular system, the integrated absorption coef- ficient of the ith fundamental infrared band is proportional to squares of the derivatives with respect to the normal coordi- nate Q_i of the permanent electric dipole moment M_␣⁽⁰⁾. Analogously, the Raman intensities depend upon the deriva- tives of the molecular electric dipole polarizability

␣

_␣␤^, while the hyper-Raman intensities are related to the deriva- tives of first hyperpolarizability

␤

_␣␤␥^. 共However, we note that, in Raman scattering, the driving electric field is nor- mally an optical field and not a static field, as considered here.兲

By means of a similarity transformation, the derivative of a given molecular property with respect to Q_i can be rewritten in terms of the geometrical derivatives with respect to the Cartesian nuclear coordinates. A straightforward appli- cation of the Hellmann–Feynman theorem to the expectation value of the molecular Hamiltonian, to various orders in the perturbing electric field, then shows that the geometrical property derivatives are related to the Lorentz force acting upon the nuclei.^1–10 These forces depend on the effective electric field experienced by the nuclei, which is the sum of the externally applied field共customarily assumed to be uni- form within the long-wavelength approximation¹¹兲and of the nonuniform field induced at the nuclei by the molecular elec- trons in response to the external perturbation.

This phenomenology is conveniently described in terms of the electric dipole shieldings and hypershieldings at the nuclei.3,4,7,8,10,12,13Thus, although the absorption frequencies in vibrational spectra are quantized, the corresponding inten- sities can be interpreted semiclassically in terms of total elec-

tric fields in the proximity of the nuclei and rationalized in terms of local effective Lorentz forces driving a change of the nuclear motion. The concept of nuclear electric shield- ings therefore provides a useful tool for explaining the elec- tronic response to the external perturbation in different parts of the molecule on the basis of the dynamics of the vibrating nuclei. For instance, hyperchromic 共hypochromic兲effects—

that is, the increase 共decrease兲in absorption intensities of a given chromophore—can be interpreted and estimated on the basis of the electron distribution in the vicinity of the vibrat- ing moiety and the changes caused by substituents in the molecule. In short, theoretical calculations of the electric shielding at the nuclei can contribute to the understanding of a number of phenomena in connection with recent studies of infrared,1,2,5,6,14,15 Raman,^{16 –19} and hyper-Raman intensities.²⁰

II. ELECTRIC FIELD AT A NUCLEUS

A. Nuclear electric shieldings and hypershieldings In the presence of an external electric perturbation, the electric field induced at nucleus I in an n-electron molecular system may be expanded in a power series in the electric field E_␣.^10,12,21In standard notation,^13,21,22we obtain

⌬

具

^EI␣n

典

⫽⫺

␥

_␣␤^{I E}_␤⫹¹2

␾

_␣,␤␥^{I E}_␤^E_␥⫹¹6

␩

␣,␤␥^I ␦E_␤E_␥E_␦

⫹¯, 共1兲

where the dimensionless tensor

␥

_␣␤^I ⫽⫺1

ប_j

兺

_⫽a

_␻

²_ja^R共

^具

^{a兩EIn␣兩j}

^典具

^j兩

^␮

^␤兩a

^典

^{兲 共2兲}

3142

0021-9606/2004/120(7)/3142/10/$22.00 © 2004 American Institute of Physics

describes the dipole electric shielding at the nucleus, while the first hypershielding is given by

␾

_␣I,␤␥⫽S共E_Iⁿ_␣,

␮

_␤^,

␮

_␥兲

冋

^ប12

^兺

^j⫽a

^␻

^⫺ja1

^冉

^k

^兺

^⫽a

^具

^{a兩EIn␣兩j}

^典

⫻

具

^j兩

␮

_␤兩k

典具

^k兩

␮

_␥兩a

典 ␻

ka⫺1⫺

具

^a兩E_Iⁿ_␣兩a

典具

^a兩

␮

_␤兩j

典

⫻

具

^j兩

␮

_␥兩a

典 ␻

⫺ja1

冊 册

^{. 共3兲}

Here the symbol S implies a sum over the six terms obtained by permuting the operators E_Iⁿ_␣,

␮

_␤^{, and}

␮

_␥^.21

In the Hellmann–Feynman approximation, we assume that the conditions of the Hellmann–Feynman theorem are satisfied. The dipole shielding tensor Eq.共2兲is then related to the geometrical derivatives of the molecular electric dipole moment

M_␣^共0兲⫽

具

^a兩

␮

_␣兩a

典

⫹_I

兺

_⫽^N₁ ^{ZIeRI␣ 共4兲}

in the following manner:^12,21

P_␣␤^I共0兲⫽ⵜI␣M_␤^共0兲⫽Z_Ie共

␦

␣␤⫺

␥

_␣␤^I 兲. 共5兲 Here the R_I␣ are the coordinates of nucleus I, ⵜI␣

⫽

⳵

^/

⳵

^RI␣, and we have introduced the permanent共intrinsic兲 atomic polar tensor with elements P_␣␤^I(0).3,10,14,23,24

A similar relation between the geometrical derivatives of the electric dipole polarizability

␣

_␣␤⫽ប^⫺1_j

兺

_⫽a ²

^␻

^⫺ja1R共

^具

^a兩

^␮

^␣兩j

^典具

^j兩

^␮

^␤兩a

^典

^{兲 共6兲}

and the nuclear electric hypershieldings is given by^10,13,21 ⵜI␣

␣

␤␥⫽Z_Ie

␾

_␣I,␤␥

, 共7兲

again in the Hellmann–Feynman approximation.

The relations Eqs.共5兲and共7兲suggest that the treatment of vibrational absorption intensities can be based on the geo- metrical derivatives of the total molecular electric dipole mo- ment in the presence of the electric perturbation

M⫽M^共0兲⫹

␣

_␣␤^E_␤⫹¹2

␤

_␣␤␥^E_␤^E_␥⫹¯ 共8兲 expressed in terms of an analogous expansion of the total atomic polar tensor

P_␣␤^I ⫽P_␣␤^I共0兲⫹P_␣␤^I _,␥E_␥⫹¹2P_␣␤^I _,␥␦E_␥E_␦⫹¯, 共9兲 where P_␣␤^I(0) is defined in Eq.共5兲and where

P_␣␤^I _,␥⫽Z_Ie

␾

_␣I,␤␥

, 共10兲

P_␣␤^I _,␥,␦⫽Z_Ie

␩

_␣^I,␤␥␦. 共11兲

For I⫽1,2,...,N, the set of total atomic polar tensors P_␣␤^I , P_␣␤^I _,␥, P_␣␤^I _,␥␦,... in Eq. 共9兲 therefore constitutes a set of molecular response properties that fully accounts for the vi- brational absorption intensities.

B. Sum rules

As can be shown from the extra-diagonal hypervirial relationships,⁴the dipole electric shielding tensor Eq.共2兲ful- fills the following three fundamental constraints^12,21

I

兺

⫽1 N

Z_I

␥

_␣␤^I ⫽n

␦

␣␤, 共12兲

I

兺

⫽1 N

⑀

␣␤␥Z_IR_I␤

␥

_␥^I_␦⫽

⑀

␣␤␦

具

^a兩R␤兩a

典

^{, 共13兲}

I

兺

⫽1 N

Z_IR_I␣

␥

_␣␤^I ⫽

具

^a兩R_␤兩a

典

⫹

兵

^Tn,R_␤

其

⫺1. 共14兲

Here R_␣ are the coordinates of the electronic centroid, T_n is the operator for the electronic kinetic energy, and we have introduced the notation

兵

^Tn,R_␣

其

⫺1⫽1

ប

兺

_j⫽a

_␻

²_ja^R共

^具

^a兩Tn兩j

^典具

^j兩R␣兩a

^典

^{兲. 共15兲}

As discussed in Refs. 12 and 21, these relations have many ramifications and embody a number of physical aspects.

First, Eq.共12兲is a Thomas–Reiche–Kuhn sum rule for oscillator strengths expressed in the mixed length- acceleration picture; also, it provides the condition for trans- lational invariance of the molecular dipole moment. Next, Eq.共13兲represents the condition for rotational invariance^12,21 and can be interpreted by invoking the relation Eq. 共5兲 be- tween nuclear shieldings and atomic polar tensors. Expressed in terms of the molecular dipole moment, it takes the form^1,2

I

兺

⫽1 N

⑀

␣␤␥R_I␤ⵜI␥M_␦^共0兲⫽

⑀

␣␤␦M_␤^共0兲, 共16兲

and provides a formal resolution of the dipole moment tensor into atomic contributions.²⁵Finally, the sum rule Eq.共14兲is obtained from the virial theorem²⁶ and accounts for scale invariance.⁴

In a similar manner, the electric hypershielding tensors Eq. 共3兲 satisfy constraints for invariance under coordinate transformations, which can be expressed as quantum me- chanical sum rules.10,21,27,28

The conditions for translational invariance of the electric dipole polarizability upon a change of the origin for the coordinate system, r⬘→r⬙⫽r⬘⫹d, are to first order in d given by

I

兺

⫽1 N

Z_I

␾

_␣I,␤␥⫽_I

兺

_⫽^N₁ ^ⵜI␣

^␣

^{␤␥⫽0. 共17兲}

Likewise, the rotational and virial conditions are obtained in the form,²¹

I

兺

⫽1 N

Z_Ie

⑀

␣␤␥R_I␤

␾

_␥I,␦⑀⫽_I

兺

_⫽^N₁

^⑀

^{␣␤␥RI␤ⵜI␥}

^␣

^␦⑀

⫽

⑀

␣␤␦

␣

␤⑀⫹

⑀

␣␤⑀

␣

_␦␤, 共18兲

I

兺

⫽1 N

Z_IeR_I␣

␾

_␣I,␤␥⫽I

兺

⫽1 N

R_I␣ⵜI␣

␣

␤␥

⫽3

␣

_␤␥⫹

兵

^Tn,

␮

_␤^,

␮

_␥

其

⫺2, 共19兲

where we have introduced

兵

^Tn,

␮

␤,

␮

␥

其

⫺2⫽S共T_n,

␮

␤,

␮

␥兲

冋

^ប12

^兺

^j⫽a

^␻

^⫺ja1

⫻

冉

^k

^兺

^⫽a

^␻

^ka⫺1

^具

^a兩Tn兩j

^典具

^j兩

^␮

^␤兩k

^典

⫻

具

^k兩

␮

␥兩a

典

^⫺

␻

ja⫺1

具

^a兩Tn共0兲兩a

典具

^a兩

␮

␤兩j

典

⫻

具

^j兩

␮

_␥兩a

典 冊 册

^共20兲

in analogy with Eq. 共15兲. We note that these sum rules are valid at all nuclear positions, not only at equilibrium, see, for instance, Ref. 27.

The existence of constraints for rototranslational invari- ance fulfilled by the Hellmann–Feynman derivatives Eqs.

共12兲,共13兲,共17兲, and 共18兲implies that, in the absence of ad- ditional conditions such as molecular symmetry, there are only 3N⫺6 (3N⫺5 for linear molecules兲linearly indepen- dent共Cartesian兲geometrical derivatives of the electric dipole moment and of the electric polarizability, and more generally of any molecular property.

TABLE I. Hartree–Fock values of electric hypershieldings at the H2O nuclei in the cc-pV5Z basis. Atomic units are used in all the tables. The geometry was optimized using the Hellmann–Feynman gradient.

␣ ␤⫽x

␥⫽x ␤⫽x

␥⫽y ␤⫽y

␥⫽y ␤⫽x

␥⫽z ␤⫽y

␥⫽z ␤⫽z

␥⫽z

␾_␣O,␤␥ x 0.000 ⫺0.451 0.000 0.000 0.000 0.000

y ⫺0.610 0.000 ⫺0.508 0.000 0.000 ⫺0.157

z 0.000 0.000 0.000 0.000 ⫺0.013 0.000

␾␣,␤␥H1 x 4.245 2.111 2.121 0.000 0.000 0.820

y 2.649 1.793 2.249 0.000 0.000 0.512

z 0.000 0.000 0.000 0.430 0.301 0.000

␾␣,␤␥H2 x ⫺4.245 2.111 ⫺2.121 0.000 0.000 ⫺0.820

y 2.649 ⫺1.793 2.249 0.000 0.000 0.512

z 0.000 0.000 0.000 ⫺0.430 0.301 0.000

兺I⫽1

N ZIe␾␣,␤␥I x 0.000 0.613 0.000 0.000 0.000 0.000

y 0.418 0.000 0.431 0.000 0.000 ⫺0.235

z 0.000 0.000 0.000 0.000 0.496 0.000

兺_I⫽1^N ⑀␣␦⑀R_I␦ZIe␾⑀,␤␥I

x 0.000 0.000 0.000 0.000 0.284 0.000

y 0.000 0.000 0.000 ⫺1.206 0.000 0.000

z 0.000 0.817 0.000 0.000 0.000 0.000

⑀␣␦␤␣␦␥⫹⑀␣␦␥␣␤␦ x 0.000 0.000 0.000 0.000 0.994 0.000

y 0.000 0.000 0.000 ⫺1.707 0.000 0.000

z 0.000 0.713 0.000 0.000 0.000 0.000

兺I⫽1

N RI␣ZIe␾_␣I,␤␥

17.45 0.00 10.60 0.00 0.00 3.62

3␣␤␥⫹兵^Tn

(0),␮␤,␮␥其⫺2 19.89 0.00 13.92 0.00 0.00 8.08

TABLE II. Hartree–Fock values of electric hypershieldings at the H2O nuclei in the aug-cc-pV5Z basis. The geometry was optimized using the Hellmann–Feynman gradient.

␣ ␤⫽x

␥⫽x

␤⫽x

␥⫽y

␤⫽y

␥⫽y

␤⫽x

␥⫽z

␤⫽y

␥⫽z

␤⫽z

␥⫽z

␾_␣O,␤␥ x 0.000 ⫺0.487 0.000 0.000 0.000 0.000

y ⫺0.653 0.000 ⫺0.525 0.000 0.000 ⫺0.154

z 0.000 0.000 0.000 0.000 ⫺0.045 0.000

␾_␣H1,␤␥ x 4.409 2.043 2.330 0.000 0.000 1.138

y 2.649 1.793 2.292 0.000 0.000 0.626

z 0.000 0.000 0.000 0.380 0.241 0.000

␾_␣H2,␤␥ x ⫺4.409 2.043 ⫺2.330 0.000 0.000 ⫺1.138

y 2.649 ⫺1.793 2.292 0.000 0.000 0.626

z 0.000 0.000 0.000 ⫺0.380 0.241 0.000

兺I⫽1

N ZIe␾␣,␤␥I x 0.000 0.190 0.000 0.000 0.000 0.000

y 0.075 0.000 0.387 0.000 0.000 0.024

z 0.000 0.000 0.000 0.000 0.121 0.000

兺I⫽1

N ⑀␣␦⑀RI␦ZIe␾⑀,␤␥I x 0.000 0.000 0.000 0.000 0.439 0.000

y 0.000 0.000 0.000 ⫺1.068 0.000 0.000

z 0.000 0.680 0.000 0.000 0.000 0.000

⑀␣␦␤␣␦␥⫹⑀␣␦␥␣␤␦ x 0.000 0.000 0.000 0.000 0.550 0.000

y 0.000 0.000 0.000 ⫺1.107 0.000 0.000

z 0.000 0.557 0.000 0.000 0.000 0.000

兺_I⫽1^N RI␣ZIe␾_␣,␤␥^I 18.17 0.00 11.30 0.00 0.00 4.56

3␣␤␥⫹兵Tn

(0),␮␤,␮␥其_⫺2 18.31 0.00 11.80 0.00 0.00 4.86

It should be noted that the sum rules Eqs.共12兲–共14兲for the shieldings and Eqs.共17兲–共19兲for the hypershieldings are exactly satisfied in all calculations based on the algebraic approximation, for all basis sets, provided the geometrical derivatives of the electric polarizability are evaluated not in the Hellmann–Feynman approximation but rather by ana- lytic procedures,¹⁵ using explicit differentiation of the elec- tronic wave function, two-electron integrals, and so on. In this non-Hellmann–Feynman approximation, Eqs.共17兲–共19兲 are useful for checking the correctness of the computer pro- gram for analytic derivatives but not for gauging the accu- racy of calculated hypershieldings.

By contrast, in the Hellmann–Feynman approximation, the constraints Eqs.共17兲–共19兲for hypershieldings Eq.共3兲are not satisfied except in the limit of a complete one-electron basis.²¹The closeness of the calculated sum rules to the exact results may then serve as an indicator for a given basis set to comply with the conditions imposed by the Hellmann–

Feynman theorem and consequently as a measure for the accuracy of calculated hypershieldings.

We finally note that, in both approaches, large basis sets are needed for accurate calculations of nuclear electric hypershieldings—either by an analytical procedure, or by the Hellmann–Feynman approach relying on Eq.共3兲.

TABLE III. Hartree–Fock values of electric hypershieldings at the H2O nuclei in the cc-pCV5Z basis. The geometry was optimized using the Hellmann–Feynman gradient.

␣ ␤⫽x

␥⫽x ␤⫽x

␥⫽y ␤⫽y

␥⫽y ␤⫽x

␥⫽z ␤⫽y

␥⫽z ␤⫽z

␥⫽z

␾_␣O,␤␥ x 0.000 ⫺0.472 0.000 0.000 0.000 0.000

y ⫺0.612 0.000 ⫺0.508 0.000 0.000 ⫺0.131

z 0.000 0.000 0.000 0.000 ⫺0.046 0.000

␾␣,␤␥H1 x 4.283 2.086 2.116 0.000 0.000 0.833

y 2.595 1.730 2.142 0.000 0.000 0.494

z 0.000 0.000 0.000 0.419 0.281 0.000

␾␣,␤␥H2 x ⫺4.283 2.086 ⫺2.116 0.000 0.000 ⫺0.833

y 2.595 ⫺1.730 2.142 0.000 0.000 0.494

z 0.000 0.000 0.000 ⫺0.419 0.281 0.000

兺I⫽1

N ZIe␾␣,␤␥I x 0.000 0.391 0.000 0.000 0.000 0.000

y 0.296 0.000 0.218 0.000 0.000 ⫺0.063

z 0.000 0.000 0.000 0.000 0.197 0.000

兺_I⫽1^N ⑀␣␦⑀R_I␦ZIe␾⑀,␤␥I

x 0.000 0.000 0.000 0.000 0.461 0.000

y 0.000 0.000 0.000 ⫺1.186 0.000 0.000

z 0.000 0.675 0.000 0.000 0.000 0.000

⑀␣␦␤␣␦␥⫹⑀␣␦␥␣␤␦ x 0.000 0.000 0.000 0.000 0.921 0.000

y 0.000 0.000 0.000 ⫺1.692 0.000 0.000

z 0.000 0.770 0.000 0.000 0.000 0.000

兺I⫽1

N RI␣ZIe␾_␣I,␤␥

17.54 0.00 10.46 0.00 0.00 3.47

3␣␤␥⫹兵^Tn

(0),␮␤,␮␥其⫺2 19.85 0.00 13.63 0.00 0.00 8.03

TABLE IV. Hartree–Fock values of electric hypershieldings at the H2O nuclei in the aug-cc-pCV5Z basis. The geometry was optimized using the Hellmann–Feynman gradient.

␣ ␤⫽x

␥⫽x

␤⫽x

␥⫽y

␤⫽y

␥⫽y

␤⫽x

␥⫽z

␤⫽y

␥⫽z

␤⫽z

␥⫽z

␾_␣O,␤␥ x 0.000 ⫺0.496 0.000 0.000 0.000 0.000

y ⫺0.645 0.000 ⫺0.534 0.000 0.000 ⫺0.148

z 0.000 0.000 0.000 0.000 ⫺0.051 0.000

␾_␣H1,␤␥ x 4.450 2.024 2.331 0.000 0.000 1.152

y 2.601 1.743 2.201 0.000 0.000 0.605

z 0.000 0.000 0.000 0.372 0.225 0.000

␾_␣H2,␤␥ x ⫺4.450 2.024 ⫺2.331 0.000 0.000 ⫺1.152

y 2.601 ⫺1.743 2.201 0.000 0.000 0.605

z 0.000 0.000 0.000 ⫺0.372 0.225 0.000

兺I⫽1

N ZIe␾␣,␤␥I x 0.000 0.081 0.000 0.000 0.000 0.000

y 0.043 0.000 0.127 0.000 0.000 0.029

z 0.000 0.000 0.000 0.000 0.038 0.000

兺I⫽1

N ⑀␣␦⑀RI␦ZIe␾⑀,␤␥I x 0.000 0.000 0.000 0.000 0.459 0.000

y 0.000 0.000 0.000 ⫺1.053 0.000 0.000

z 0.000 0.622 0.000 0.000 0.000 0.000

⑀␣␦␤␣␦␥⫹⑀␣␦␥␣␤␦ x 0.000 0.000 0.000 0.000 0.504 0.000

y 0.000 0.000 0.000 ⫺1.092 0.000 0.000

z 0.000 0.588 0.000 0.000 0.000 0.000

兺_I⫽1^N RI␣ZIe␾_␣,␤␥^I 18.19 0.00 11.26 0.00 0.00 4.55

3␣␤␥⫹兵Tn

(0),␮␤,␮␥其_⫺2 18.29 0.00 11.56 0.00 0.00 4.84

III. NUCLEAR ELECTRIC HYPERSHIELDINGS

We report in this section calculations of nuclear electric hypershieldings in H₂O, CO, N₂, and NH₃. The purpose of these calculations is to explore the usefulness of the Hellmann–Feynman approximation as a computational tool in quantum chemistry. In particular, we would like to estab- lish the basis-set requirements for electric hypershieldings in polyatomic molecules.

Studies of the basis set dependence of the computed components

␾

x,xz

I of the hypershielding tensor for diatomics, giving the electric field at the nuclei in the x direction, per- pendicular to the internuclear axis, and proportional to the product E_xE_z of the electric field strengths applied parallel and perpendicular to the internuclear axis, showed that ex- tended basis sets are necessary for near Hartree–Fock accuracy.^13,28

A. Computational details

As is well known, the requirements that must be met for the Hellmann–Feynman theorem to be applicable are quite severe.^15,29Previous attempts at developing atomic basis sets suitable for the calculation of nuclear electric shieldings³⁰ were, to some extent, successful.^31,32However, in this paper, we take a somewhat different approach to calculations in the Hellmann–Feynman approximation, exploring the use of hi- erarchical basis sets—in particular, the standard and ex- tended correlation-consistent sets of Dunning and co-workers^33–35 共cc-pVXZ, aug-cc-pVXZ, and cc-pCVXZ, aug-cc-pCVXZ兲 and the atomic natural orbital共ANO兲 basis sets of Widmark et al.³⁶ In addition, the R12 basis sets de- veloped to satisfy the completeness constraints in explicitly correlated R12 calculations are explored. As shown by Bakken et al. for forces and frequencies, these basis sets are well suited to the calculation of molecular properties in the Hellmann–Feynman approximation.³⁷

For the sake of consistency, molecular geometries em- ployed in the present ab initio study correspond to those optimized in Ref. 37 by imposing the constraint that the Hellmann–Feynman forces vanish at the Hartree–Fock level. The smaller correlation-consistent series, namely, the double- and triple-zeta basis sets, are not used since, for these basis sets, the search for an optimal Hellmann–

Feynman geometry usually does not converge.

The primitive Gaussian functions of the ANO (14s9 p4d3 f /8s4 p3d) and R12 (13s8 p6d5 f /7s5 p4d) basis sets were allowed to vary freely. The augmented versions of the same sets, that is, the ANO⫹ (16s11p4d3 f /10s4 p3d) and the R12⫹共15s10p6d5f/

9s5p4d兲, were obtained as described by Bakken et al.³⁷

B. Calculated properties

The polarizabilities

␣

␤␥, the nuclear electric shieldings

␥

_␣␤^I , the nuclear hypershieldings

␾

_␣I,␤␥

, and the auxiliary tensors

兵

^Tn,R_␣

其

⫺1and

兵

^Tn,

␮

_␤^,

␮

_␥

其

⫺2of Eqs.共15兲and共19兲 were determined at the Hartree–Fock level of theory, using the DALTON program³⁸ 共see Tables I–X兲. For selected basis sets, we report: the Cartesian components of the nuclear elec- tric hypershielding of the nuclei, the left-hand side of the translational sum rule Eq.共17兲, the left- and right-hand sides of the rotational sum rule Eq. 共18兲, the left- and right-hand sides of the virial sum rule Eq.共19兲.

Complete results for the water molecule are shown for six of the twelve basis sets examined, results for the remain- ing six sets can be found in the supplementary material.³⁹ However, a comparison of all twelve basis sets is given in Table VII, where performance is measured by the root-mean- square values of three entities that become exactly zero in the limit of a complete basis set. Finally, we provide full results

TABLE V. Hartree–Fock values of electric hypershieldings at the H2O nuclei in the R12 basis. The geometry was optimized using the Hellmann–Feynman gradient.

␣ ␤⫽x

␥⫽x ␤⫽x

␥⫽y ␤⫽y

␥⫽y ␤⫽x

␥⫽z ␤⫽y

␥⫽z ␤⫽z

␥⫽z

␾_␣O,␤␥ x 0.000 ⫺0.498 0.000 0.000 0.000 0.000

y ⫺0.641 0.000 ⫺0.532 0.000 0.000 ⫺0.150

z 0.000 0.000 0.000 0.000 ⫺0.049 0.000

␾␣,␤␥H1 x 4.482 2.019 2.336 0.000 0.000 1.159

y 2.586 1.723 2.159 0.000 0.000 0.590

z 0.000 0.000 0.000 0.371 0.220 0.000

␾␣,␤␥H2 x ⫺4.482 2.019 ⫺2.336 0.000 0.000 ⫺1.159

y 2.586 ⫺1.723 2.159 0.000 0.000 0.590

z 0.000 0.000 0.000 ⫺0.371 0.220 0.000

兺I⫽1

N ZIe␾␣,␤␥I x 0.000 0.054 0.000 0.000 0.000 0.000

y 0.040 0.000 0.061 0.000 0.000 ⫺0.017

z 0.000 0.000 0.000 0.000 0.048 0.000

兺_I⫽1^N ⑀␣␦⑀R_I␦ZIe␾⑀,␤␥I

x 0.000 0.000 0.000 0.000 0.435 0.000

y 0.000 0.000 0.000 ⫺1.055 0.000 0.000

z 0.000 0.625 0.000 0.000 0.000 0.000

⑀␣␦␤␣␦␥⫹⑀␣␦␥␣␤␦ x 0.000 0.000 0.000 0.000 0.482 0.000

y 0.000 0.000 0.000 ⫺1.084 0.000 0.000

z 0.000 0.602 0.000 0.000 0.000 0.000

兺I⫽1

N RI␣ZIe␾_␣I,␤␥

18.26 0.00 11.22 0.00 0.00 4.57

3␣␤␥⫹兵^Tn

(0),␮␤,␮␥其⫺2 18.33 0.00 11.43 0.00 0.00 4.87

for three other molecules, CO, N₂, and NH₃, using the R12 basis set共see Tables VIII–X兲.

All quantities are reported in atomic units. Since the physical dimension of the

␾

_␣I,␤␥

tensor is the inverse of an electric field, we find that the conversion factor to SI units is ea₀/E_h⬇1.944 690 57⫻10^⫺12mV^⫺1, using the CODATA recommended values from Ref. 40. For electric field strengths normally attainable in the laboratory 共about 10⁷V m^⫺1), the contribution to second order in E_␣ is, ac- cording to Eq. 共1兲, about 10²V m^⫺1 for the water molecule, where

␾

_␣,␤␥^I is about one atomic unit. Since the dimension- less

␥

_␣␤^I tensor is close to one共see Tables XI and XII兲, the ratio of the first to second-order terms is about 10⁵, that is, the series Eq.共1兲converges rapidly.

C. Correlation-consistent basis sets

The results obtained with the correlation-consistent hier- archical basis sets display clearly the convergence pattern of the calculations carried out in the present work. As expected, very large basis sets are needed to achieve rototranslational and scale invariance, that is, to achieve the conditions for near Hartree–Fock accuracy of the computed nuclear electric hypershieldings. Taking the results from the largest correlation-consistent basis aug-cc-pCV5Z 共341 CGTOs兲 in Table IV as a benchmark, we note that the convergence of the sum rules, that is, the accuracy of the computed hyper- shieldings, can be controlled in two ways.

First, for a given cardinal number X, the addition of core functions improves the accuracy of the left-hand sides of the translational, rotational, and virial sum rules; second, aug- mentation with diffuse functions yields more accurate results for the electric polarizability and consequently for the right- hand side of the rotational condition of the virial sum rule.

As the latter procedure increases the basis set faster than the former, it leads more rapidly to saturation of the expansion and consequently to generally better results.

Whereas core augmentation provides a more flexible and accurate description of the core region, it also leads to a poorer description of the outer valence region, reducing the quality of the calculated polarizability—compare, for ex- ample, the results obtained with the cc-pV5Z basis共201 CG- TOs兲in Table I with those obtained with the cc-pCV5Z basis 共255 CGTOs兲 in Table III and the aug-cc-pV5Z 共287 CG- TOs兲basis in Table II.

The standard quintuple-zeta basis set gives rather poor estimates of the sum rules, with an average difference of about 0.5 a.u. between the left- and right-hand sides of Eqs.

共17兲and共18兲; even poorer results are obtained for the virial sum rule Eq. 共19兲. The addition of tight functions does not

TABLE VI. Hartree–Fock values of electric hypershieldings at the H2O nuclei in the R12⫹basis. The geom- etry was optimized using the Hellmann–Feynman gradient.

␣ ␤⫽x

␥⫽x ␤⫽x

␥⫽y ␤⫽y

␥⫽y ␤⫽x

␥⫽z ␤⫽y

␥⫽z ␤⫽z

␥⫽z

␾_␣O,␤␥ x 0.000 ⫺0.500 0.000 0.000 0.000 0.000

y ⫺0.640 0.000 ⫺0.532 0.000 0.000 ⫺0.148

z 0.000 0.000 0.000 0.000 ⫺0.051 0.000

␾␣,␤␥H1 x 4.492 2.014 2.339 0.000 0.000 1.169

y 2.573 1.711 2.138 0.000 0.000 0.586

z 0.000 0.000 0.000 0.369 0.216 0.000

␾␣,␤␥H2 x ⫺4.492 2.014 ⫺2.339 0.000 0.000 ⫺1.169

y 2.573 ⫺1.711 2.138 0.000 0.000 0.586

z 0.000 0.000 0.000 ⫺0.369 0.216 0.000

兺I⫽1

N ZIe␾␣,␤␥I x 0.000 0.024 0.000 0.000 0.000 0.000

y 0.026 0.000 0.022 0.000 0.000 ⫺0.015

z 0.000 0.000 0.000 0.000 0.024 0.000

兺_I⫽1^N ⑀␣␦⑀R_I␦ZIe␾⑀,␤␥I

x 0.000 0.000 0.000 0.000 0.443 0.000

y 0.000 0.000 0.000 ⫺1.051 0.000 0.000

z 0.000 0.609 0.000 0.000 0.000 0.000

⑀␣␦␤␣␦␥⫹⑀␣␦␥␣␤␦ x 0.000 0.000 0.000 0.000 0.463 0.000

y 0.000 0.000 0.000 ⫺1.068 0.000 0.000

z 0.000 0.605 0.000 0.000 0.000 0.000

兺I⫽1

N RI␣ZIe␾_␣I,␤␥

18.26 0.00 11.20 0.00 0.00 4.59

3␣␤␥⫹兵^Tn

(0),␮␤,␮␥其⫺2 18.35 0.00 11.35 0.00 0.00 4.73

TABLE VII. The performance of various basis sets when calculating Hartree–Fock values of electric hypershieldings at the H2O nuclei. The values shown are the root-mean-square values of all nonzero values of 兺I⫽1

N ZIe␾_␣I,␤␥

and the two differences ⌬1⫽兺I⫽1

N ⑀␣␦⑀R_I␦ZIe␾_⑀I,␤␥

⫺(⑀␣␦␤␣␦␥⫹⑀␣␦␥␣␤␦) and ⌬2⫽兺I⫽1N

RI␣ZIe␾_␣,␤␥^I ⫺(3␣_␤␥

⫹兵Tn

(0),␮␤,␮␥其2)2. Basis 兺I⫽1

N ZIe␾␣,␤␥I ⌬1 ⌬2 Average

cc-pVQZ 1.210 0.953 6.111 2.758

cc-pV5Z 0.456 0.505 3.506 1.489

cc-pCVQZ 0.475 0.615 5.353 2.148

cc-pCV5Z 0.257 0.399 3.473 1.376

aug-cc-pVQZ 1.392 0.923 1.621 1.312

aug-cc-pV5Z 0.203 0.098 0.346 0.216

aug-cc-pCVQZ 0.257 0.146 0.582 0.328

aug-cc-pCV5Z 0.073 0.040 0.248 0.120

ANO 0.155 0.049 0.610 0.272

ANO⫹ 0.102 0.017 0.465 0.195

R12 0.047 0.035 0.215 0.099

R12⫹ 0.023 0.015 0.129 0.056