Direct perturbation theory of magnetic properties and relativistic corrections for the point nuclear and Gaussian nuclear models

Direct perturbation theory of magnetic properties and relativistic corrections for the point nuclear and Gaussian nuclear models

Alf C. Hennum^a)and Wim Klopper

Theoretical Chemistry Group, Debye Institute, Utrecht University, P.O. Box 80052, NL-3508 TB Utrecht, The Netherlands

Trygve Helgaker

Department of Chemistry, University of Oslo, P.O. Box 1033 Blindern, N-0315 Oslo, Norway 共Received 4 January 2001; accepted 1 August 2001兲

Starting from the Le´vy-Leblond equation, which is the four-component nonrelativistic limit of the Dirac equation, a direct perturbation theory of magnetic properties and relativistic corrections is developed and implemented for point-charge and finite nuclei. The perturbed small components are regularized by projecting them onto an auxiliary small-component basis of Gaussian functions. The relevant operators and matrix elements are derived for the point-nuclear and Gaussian nuclear models. It is demonstrated how the usual paramagnetic spin-orbit, Fermi-contact, and spin-dipole integrals of Ramsey’s theory can be evaluated in the same manner as field and field-gradient integrals—that is, as derivatives of potential-energy integrals. A few illustrative calculations are performed. © 2001 American Institute of Physics. 关DOI: 10.1063/1.1405009兴

I. INTRODUCTION

On various occasions, the Le´vy-Leblond equation¹ has served as a zeroth-order equation for a direct perturbation theory 共DPT兲 of relativistic effects.^2,3 Direct perturbation theory follows from introducing a change in the metric into the Dirac共–Coulomb兲 equation and expanding this equation in powers of 1/c, where c is the velocity of light (c

⫽137.035 989 5 a₀E_h/ប). In Ref. 4, for example, it is dis- cussed how a four-component framework 共and correspond- ing computer program兲can be utilized to perform nonrela- tivistic calculations in the Le´vy-Leblond formalism, and how the Dirac- and Le´vy-Leblond equations are both related to the zeroth-order regular approximation 共ZORA兲 共see also Ref. 5兲.

In this paper, we shall apply a DPT of relativistic correc- tions to magnetic properties. Although the necessary theory has already been put forward by Kutzelnigg,⁶ the required matrix elements have not yet been computed. In the present paper, we propose a general procedure by which these matrix elements can be evaluated—namely, by projecting the per- turbed small components onto an appropriate basis of Gauss- ian functions. Much of the material presented in this paper is contained in Ref. 7.

Four-component relativistic calculations usually employ the Gaussian nuclear model for energy calculations; in such calculations, it is thus desirable to be able to compute nucleus-dependent molecular properties such as electric field-gradients⁸ and magnetic terms^9,10 for the Gaussian nuclear model. Accordingly, in this paper, we not only pro- vide formulas for the computation of matrix elements for the point nuclear model but also for the Gaussian nuclear model, presenting simple formulas according to which the relevant

integrals may be calculated as derivatives of the potential- energy integrals for Gaussian nuclei. In particular, the inte- grals of the paramagnetic spin-orbit 共PSO兲, Fermi-contact 共FC兲, and spin-dipole共SD兲integrals of Ramsey’s theory are obtained in the same manner as the usual electric field共EF兲 and electric field-gradient 共EFG兲integrals.

II. DPT OF MAGNETIC AND RELATIVISTIC CORRECTIONS

A. The zeroth-order Le´vy-Leblond equation

The zeroth-order starting point for our discussion is the Le´vy-Leblond equation

D⁰⁰⌿⁰⁰⫽E⁰⁰S⁰⁰⌿⁰⁰, 共1兲

D⁰⁰⫽

冉

^{␴V"p ␴⫺"2p}

冊

^{, S00⫽}

冉

^{10 00}

冊

^{, ⌿00⫽}

^冉

^␸␹0000

^冊

^.

共2兲 Here V is the nuclear potential, p is the momentum vector, and the vector ␴contains the three Pauli spin matrices,

␴x⫽

冉

^{01 10}

冊

^{, ␴y⫽}

冉

^{0i ⫺0i}

冊

^{, ␴z⫽}

冉

^{10 ⫺01}

冊

^{. 共3兲}

The two-component functions ␸^{00 and} ␹⁰⁰ are the ‘‘large’’

and ‘‘small’’ components, respectively, of the four- component spinor ⌿⁰⁰. It follows from Eq. 共1兲 that the zeroth-order small component is given by

␹⁰⁰⫽¹2␴"p␸^00, 共4兲

and substitution of this relationship into Eq. 共1兲 yields the one-electron Schro¨dinger equation

关V⫹¹2共␴"p兲共␴"p兲兴␸⁰⁰⫽共V⫹T兲␸⁰⁰⫽H␸⁰⁰⫽E⁰⁰␸^00.

共5兲

a兲Present address: Department of Chemistry, University of Oslo, P.O. Box 1033 Blindern, N-0315 Oslo, Norway.

7356

0021-9606/2001/115(16)/7356/8/$18.00 © 2001 American Institute of Physics

The unperturbed large component is expanded in a basis set 兵␰^k其of ordinary共but two-component兲atomic orbitals共AO’s兲

␸⁰⁰⫽

兺

k

c_k⁰⁰␰k, 共6兲

which we shall take to be Gaussian functions.

B. The perturbed Le´vy-Leblond equation

We now introduce two perturbations. The first is due to a magnetic field represented by the vector potential A共we as- sume that A satisfies the Coulomb gauge“"A⫽0兲,

D⁰¹⫽

冉

^{␴0"A ␴0"A}

冊

^{, S01⫽0, 共7兲}

and the second perturbation introduces relativistic correc- tions,

D²⁰⫽

冉

^{00 V0}

冊

^{, S20⫽}

冉

^{00 01}

冊

^{. 共8兲}

The traditional equations of perturbation theory can be de- rived in the usual manner by expanding the energy and the wave function in powers of the perturbation parameters and collecting equal powers, assuming the intermediate normal- ization

具⌿^mn兩S⁰⁰兩⌿⁰⁰典⫽␦0m␦0n. 共9兲

In keeping with the notation of Kutzelnigg^3,6the first super- script in Eqs.共7兲–共9兲refers to the order of c^⫺1 in the rela- tivistic perturbation and the second to the order of the mag- netic perturbation. The first-order relativistic correction is thus of order c^⫺2.

To first order, we obtain the following coupled equations for the magnetic large and small components ␸^{01 and} ␹^01, respectively:

共V⫺E⁰⁰兲␸⁰¹⫹␴"p␹⁰¹⫺E⁰¹␸⁰⁰⫹␴"A␹⁰⁰⫽0, 共10兲

␹⁰¹⫽¹2␴"p␸⁰¹⫹¹2␴"A␸^00. 共11兲

Analogously, we obtain for the relativistic perturbation

共V⫺E⁰⁰兲␸²⁰⫹␴"p␹²⁰⫺E²⁰␸⁰⁰⫽0, 共12兲

␹²⁰⫽¹2␴"p␸²⁰⫹¹4共V⫺E⁰⁰兲␴"p␸^00. 共13兲

At this point, we could proceed by inserting the small- component Eqs.共11兲and共13兲into the large-component Eqs.

共10兲 and 共12兲, respectively, and by solving the large- component equations as usual by expanding the perturbed large components in a basis of ordinary Gaussian basis func- tions. Once algebraic approximations to ␸^{01 and} ␸^{20 have} been computed, the perturbed small components are then de- termined from Eqs. 共11兲and共13兲.

However, concerning point-charge nuclei and large com- ponents that are expanded in a basis of Gaussian functions that are regular at these nuclei, the direct use of Eqs.共11兲and 共13兲leads to singularities at the nuclei in the perturbed small components共since the small components contain products of the singular operators V or ␴"A with Gaussian basis functions兲.³As we shall see, these singularities are avoided by projecting the small component Eqs.共11兲and共13兲onto an

auxiliary small-component basis 兵␨^k其. In this manner, the projected perturbed small components are regularized as well, becoming analytic at the nuclei. The same regularized large and small components occur in a formulation of DPT in the framework of stationary perturbation theory 共stationary DPT兲.³

C. Projection onto the small-component basis

A unique feature of stationary DPT³is that we do not use the perturbed small components as given by Eqs. 共11兲 and 共13兲, but rather their projections␹_P^{01 and}␹_P²⁰ onto the small component basis,

␹_P⁰¹⫽P␹^01, 共14兲

␹_P²⁰⫽P␹^20. 共15兲

Here the projector onto the small-component basis 兵␨^k其 ^is defined as

P⫽

兺

_m,n ^{兩␨m典共S⫺1兲mn具␨n兩, 共16兲}

and S^⫺1 is the inverse of the small-component overlap ma- trix S with elements S_mn⫽具␨m兩␨n典. It is important to realize that these projections introduce approximations into the theory, depending on our choice of small-component basis 兵␨k其. In principle, we could choose the small-component ba- sis in such a manner that the errors due to the approximations become small or even vanish exactly. However, we shall here assume that the projections Eqs. 共14兲and共15兲indeed intro- duce approximations and instead consider the effect of these approximations, in particular how it leads to a nonexact evaluation of certain integrals by means of the resolution-of- identity共RI兲approximation in the small-component basis.

In high-order relativistic DPT, it is common practice to choose the small-component auxiliary basis as 兵␨^k其

⫽兵␴^"p␰k其^.11–13 Then, for each function ␰k in the large-

component basis, there is exactly one function␨k⫽␴"p␰kin the small-component basis. With this auxiliary small- component basis, it follows that

具␰k兩␴"p␹_P⁰¹⫽具␰k兩␴"p␹^01, 共17兲

具␰k兩␴"p␹_P²⁰⫽具␰k兩␴"p␹^20, 共18兲

as the small-component basis contains the range of the op- erator ␴"p on the domain兵␰k其—see, for example, Ref. 14.

Thus, for calculations in the Gaussian basis兵␰k其, the projec- tion onto␨k⫽␴"p␰kyields approximate first-order perturbed small components 关Eqs.共11兲and共13兲兴but has no effect on the perturbed large components关Eqs.共10兲and共12兲兴. Clearly, the usefulness of this approach depends critically on the flex- ibility of the generated small-component basis.

Kutzelnigg⁶ has pointed out that the small-component auxiliary basis 兵␨^k其^⫽兵␴^"p␰k其 is inadequate for the expan- sion of ␹⁰¹, suggesting to include兵␴"A␰k其 in the basis. As the auxiliary basis would then contain the range of the op- erator ␴"A as well as the range of ␴"p, such an approach would lead to a compact representation of the small-

component wave function and give a balanced representation of the electronic system in the presence of relativistic and magnetic perturbations.

Nevertheless, for a practical implementation of relativis- tic corrections to magnetic properties within the framework of DPT, it is convenient to have a single small-component basis for ␹^{20 and} ␹⁰¹. This basis should be chosen as

兵␨k其傻兵␴^"p␰k其 to satisfy Eqs. 共17兲 and 共18兲 and be large

enough to yield reasonable approximations to ␹^{01. In par-} ticular, for the accurate calculation of properties, AO’s of symmetries different from those needed for the calculation of the unperturbed energy are sometimes needed. The relative advantages and disadvantages of the two approaches共the use of one or several small-component bases兲 will not be dis- cussed further here; in the following, we shall instead con- sider the effect of the RI approximation Eqs.共14兲and共15兲on the first- and second-order energies.

D. First-order corrections

For the first-order magnetic and relativistic energies, we obtain

E⁰¹⫽具␸⁰⁰兩␴"A兩␹⁰⁰典^⫹具␹⁰⁰兩␴"A兩␸⁰⁰典^{, 共19兲}

E²⁰⫽具␹⁰⁰兩V⫺E⁰⁰兩␹⁰⁰典^. 共20兲 Obviously, these energies are not affected by the projection onto the small-component basis. Using Eq.共4兲, we find that the first-order energies may be expressed entirely in terms of the zeroth-order large components as

E⁰¹⫽¹2具␸⁰⁰兩兵␴^"A,␴"p其⫹兩␸⁰⁰典^, 共21兲

E²⁰⫽¹4具␸⁰⁰兩␸"pV␴"p兩␸⁰⁰典⫺¹2E⁰⁰具␸⁰⁰兩T兩␸⁰⁰典^, 共22兲

where we have introduced the anticommutator

兵␴^"A,␴"p其⫹⫽共␴"A兲共␴"p兲⫹共␴"p兲共␴"A兲. 共23兲

We shall return to the evaluation of integrals over this and similar operators in Sec. III, when we consider the magnetic perturbation from finite and point-charge nuclei.

E. Second-order corrections

To second order in the magnetic and relativistic pertur- bations, we obtain

E⁰²⫽具␸⁰⁰兩␴"A兩␹_P⁰¹典⫹具␹⁰⁰兩␴"A兩␸⁰¹典^, 共24兲

E⁴⁰⫽具␹⁰⁰兩V⫺E⁰⁰兩␹_P²⁰典⫺E²⁰具␹⁰⁰兩␹⁰⁰典^, 共25兲

E²¹⫽2具␸⁰⁰兩␴"A兩␹_P²⁰典⫹2具␹⁰⁰兩␴"A兩␸²⁰典⫺E⁰¹具␹⁰⁰兩␹⁰⁰典

共26兲

⫽2具␹⁰⁰兩V⫺E⁰⁰兩␹_P⁰¹典⫺E⁰¹具␹⁰⁰兩␹⁰⁰典^. 共27兲 Thus, the second-order energies do depend on the RI ap- proximation, containing the perturbed small-component functions linearly in situations different from Eqs.共17兲 and 共18兲. In particular, we might worry that the two alternative expressions for the mixed magnetic-relativistic corrections 关Eqs.共26兲and共27兲兴are not equivalent in the presence of the projection. To verify their equivalence, we must establish that the Dalgarno–Stewart interchange theorem of double-

perturbation theory¹⁵holds not only for the exact small com- ponents␹^01and␹²⁰but also for their projected counterparts

␹_P^{01, and}␹_P^20,

具␸⁰⁰兩␴"A兩␹_P²⁰典^⫹具␹⁰⁰兩␴"A兩␸²⁰典^⫽具␹⁰⁰兩V⫺E⁰⁰兩␹_P⁰¹典^.

共28兲 The two sides of Eq. 共28兲can be rewritten as

具␸⁰⁰兩␴"A兩␹_P²⁰典⫹具␹⁰⁰兩␴"A兩␸²⁰典

⫽¹2具␸⁰⁰兩兵␴^"A,␴"p其⫹兩␸²⁰典

⫹¹4具␸⁰⁰兩␴"AP共V⫺E⁰⁰兲␴"p兩␸⁰⁰典^, 共29兲

具␹⁰⁰兩V⫺E⁰⁰兩␹_P⁰¹典⫽¹4具␸⁰⁰兩␴"p共V⫺E⁰⁰兲␴"p兩␸⁰¹典

⫹¹4具␸⁰⁰兩␸"p共V⫺E⁰⁰兲P␴"A兩␸⁰⁰典^{. 共30兲}

The second terms on the right-hand sides of Eqs. 共29兲 and 共30兲are expectation values, with the projectorPacting as an RI between␴"p(V⫺E⁰⁰) and␴"A. The equivalence of these terms is immediately obvious, and it is also easily verified that the first terms on the right-hand sides yield the same energy contribution.

To see more clearly the effect of the projection on the second-order energies, let us rewrite these exclusively in terms of the large components,

E⁰²⫽¹2具␸⁰⁰兩␴"AP␴"A兩␸⁰⁰典⫹¹2具␸⁰⁰兩兵␸^"A,␴"p其⫹兩␸⁰¹典^,

共31兲

E⁴⁰⫽¹8具␸⁰⁰兩␴"p共V⫺E⁰⁰兲P共V⫺E⁰⁰兲␴"p兩␸⁰⁰典

⫺¹2E²⁰具␸⁰⁰兩T兩␸⁰⁰典^⫹14具␸⁰⁰兩␸"p共V⫺E⁰⁰兲␴"p兩␸²⁰典^,

共32兲

E²¹⫽¹2具␸⁰⁰兩␴"AP共V⫺E⁰⁰兲␴"p兩␸⁰⁰典^⫺12E01具␸⁰⁰兩T兩␸⁰⁰典

⫹具␸⁰⁰兩兵␴"A,␴"p其⫹兩␸²⁰典^, 共33兲

⫽¹2具␸⁰⁰兩␴"AP共V⫺E⁰⁰兲␴"p兩␸⁰⁰典⫺¹2E⁰¹具␸⁰⁰兩T兩␸⁰⁰典

⫹¹2具␸⁰⁰兩␴"p共V⫺E⁰⁰兲␴"p兩␸⁰¹典^{. 共34兲}

We have here written the second-order energies with the ex- pectation 共diamagnetic兲 term first and the relaxation 共para- magnetic兲 term second. Clearly, the projection affects only the expectation value. For the magnetic second-order energy, for example, the only consequence of the DPT approach is that we are now dealing with the operator ␴"AP␴"A rather than the operator (␴"A)(␴"A)⫽A² that occurs in the stan- dard nonrelativistic Ramsey formulation. For computations of magnetizabilities, nuclear shieldings, and indirect spin- spin couplings, this means that the diamagnetic terms are evaluated by means of the RI, whereas the paramagnetic terms are unaffected by the projector and thus the same as in the Ramsey theory. In the approach of Kutzelnigg,⁶functions of the form 兵␴"A␰k其 are included in the auxiliary basis.

Since the auxiliary basis then contains the range of the op- erator␴"A, this approach would indeed lead to diamagnetic terms of the form A². In passing, we note that expressions for higher-order energies can be found in Ref. 6.

III. DPT MATRIX ELEMENTS A. Nuclear magnetic dipole field

In the present section, we consider the magnetic vector potential and the magnetic induction set up by the dipole moment ␮kassociated with a finite nuclear charge distribu- tion. For a point-charge nucleus at K, the vector potential at the position r of the electron is given by

A共r,K兲⫽␮kÃ共r⫺K兲

兩r⫺K兩³ . 共35兲

For a nucleus described by a normalized Gaussian distribu- tion at K,

G_␩共R_k兲⫽

冉

^␲␩

冊

^{3/2e⫺␩Rk2, 共36兲}

with

R_k⫽兩R⫺K兩, 共37兲

the vector potential is obtained by integrating the potential 关Eq. 共35兲兴over this distribution:

A_k共r兲⫽

冕

^{A共r,R兲G␩共Rk兲dR 共38兲}

⫽⫺␮k⫻“

冕

^G兩r^␩⫺^共RR^k兩^兲dR, 共39兲 where“is the gradient operator with respect to the coordi- nates of the electron r. The integral in Eq.共39兲may be evalu- ated as

Vk⫽

冕

^G兩r^␩⫺^共RR^k兩^兲dR⫽P共¹2,␩^rk 2兲

r_k , 共40兲

where

P共a,z兲⫽ 1

⌫共a兲

冕

0 z

t^a⫺1e^⫺tdt 共a⬎0兲 共41兲 is the incomplete gamma function and r_k the distance from the electron to nucleus k,

r_k⫽兩r⫺K兩. 共42兲

We may then write the nuclear vector potential and the in- duction associated with the dipole moment␮kas

A_k⫽␮kÓkVk, 共43兲

B_k⫽“ÃA_k⫽共␮k"“^k兲“^kVk⫺␮kⵜk

2Vk, 共44兲

where“kis the gradient operator with respect to the position of the nucleus K.

The expressions 共43兲 and 共44兲 are general in that they hold for any spherical nuclear distribution, not just the Gaussian distribution—provided, of course, that the integra- tion in Eq. 共40兲 is carried over the appropriate distribution.

Moreover, as we shall see shortly, these expressions are in a form that is perfectly suitable for integration over Gaussian orbitals. Still, it is instructive to compare Eqs.共43兲and共44兲 with the corresponding expressions for point charge nuclei.

In terms of the incomplete gamma function, we obtain the

following expressions for the electrostatic potential, mag- netic vector potential, and the magnetic induction of a finite Gaussian-shaped nucleus:

A_k^␩⫽P共³2,␩^rk 2兲␮kÃr_k

r_k³ , 共45兲

B_k^␩⫽8␲

3 G_␩共r_k兲␮k⫹P共⁵2,␩^rk

2兲3共␮k•r_k兲r_k⫺r_k²␮k

r_k⁵ . 共46兲 The corresponding expressions for point-charge nuclei are given by

A_k^pnt⫽␮kÃr_k

r_k³ , 共47兲

B_k^pnt⫽8␲

3 ␦共r_k兲␮k⫹3共␮k"r_k兲r_k⫺r_k²␮k

r_k⁵ . 共48兲

These expressions may be obtained either by letting ␩ ⁱⁿ Eqs. 共45兲and共46兲tend to infinity or by differentiating Eqs.

共43兲 and共44兲with Vk⫽1/r_k. The expressions for finite nu- clei differ from those for point-charge nuclei by the presence of the incomplete gamma functions and the substitution of the Dirac delta function by a Gaussian distribution. As the Gaussian exponent␩increases, the incomplete gamma func- tions tend to unity and the Gaussian distribution becomes the Dirac delta function.

Although it is not our main concern here, we note that the nuclear electrostatic potential and the associated EF and EFG operators 共arising from nuclear displacements兲may be treated in the same manner when point-charge nuclei are replaced by Gaussian distributions. Thus, using the function 共40兲, we may write the general expressions for these opera- tors in the form

V_k^␩⫽⫺Z_kVk␩⫽⫺P共¹2,␩^rk 2兲Z_k

r_k, 共49兲

E_k^␩⫽Z_k“^kVk␩⫽⫺P共³2,␩^rk 2兲Z_kr_k

r_k³, 共50兲 F_k^␩⫽⫺Z_k“k“k

TVk␩

⫽4␲

3 Z_kG_␩共r_k兲I₃⫺P共⁵2,␩^rk

2兲Z_k3r_kr_k^T⫺r_k²I₃

r_k⁵ , 共51兲 where I₃ is the three-by-three unit matrix. Again, the intro- duction of finite nuclear distributions is taken care of by the introduction of incomplete gamma functions and the use of a Gaussian distribution rather than a Dirac function for the contact term.

B. Integrals

Having developed convenient expressions for the vector potential and magnetic induction共43兲and共44兲, we shall con- sider the evaluation of integrals involving these functions. In particular, we are interested in the matrix elements of the operators

共␴"A_k兲共␴"p兲⫽A_k"p⫹i␴"A_k⫻p, 共52兲

共␴"p兲共␴"A_k兲⫽A_k"p⫹i␴"pÃA_k, 共53兲

1

2兵␴^"Ak,␴"p其⫹⫽A_k"p⫹B_k"s, 共54兲

where we have assumed that the vector potential is diver- genceless and introduced the spin of the electron

s⫽¹2␴^. 共55兲

Let us assume that ␸a and␸b are two real Gaussians cen- tered at positionsAandB, respectively. Inserting the expres- sions共43兲and共44兲in共53兲–共54兲, we obtain

具^a兩共␴"A_k兲共␴"p兲兩b典⫽i␮k"“kÓb具^a兩Vk兩b典

⫺共␮k"“b兲共“k"␴兲具^a兩Vk兩b典

⫹共␮k"␴兲共“b"“k兲具^a兩Vk兩b典^, 共56兲

具^a兩共␴"p兲共␴"A_k兲兩b典^⫽i␮k"“aÓk具^a兩Vk兩b典

⫺共␮k"“a兲共“k"␴兲具^a兩Vk兩b典

⫹共␮k"␴兲共“a"“k兲具^a兩Vk兩b典^{, 共57兲}

具^a兩共A_k"p⫹B_k"s兩b典⫽i␮k"“kÓb具^a兩Vk兩b典

⫹¹2共␮k"“^k兲共“^k"␴兲具^a兩Vk兩b典

⫺¹2共␮k"␴兲共“k"“k兲具^a兩Vk兩b典^. 共58兲

For a more compact representation, we introduce the nota- tion

V_kab^m⫻n⫽“mÓn具^a兩Vk兩b典^, 共59兲 V_kab^mn⫽“^m“n

T具^a兩Vk兩b典^{. 共60兲}

The matrix elements may now be written in the form

具^a兩共␴"Ak兲共␴"p兲兩b典⫽i␮k"V_kab^k⫻b⫺␮k

T共V_kab^bk ⫺I₃TrV_kab^bk 兲␴^, 共61兲

具^a兩共␴"p兲共␴"A_k兲兩b典^⫽i␮k"V_kab^a⫻k⫺␮k

T共V_kab^ak ⫺I₃TrV_kab^ak 兲␴^, 共62兲

具^a兩A_k"p⫹B_k•s兩b典⫽i␮k•V_kab^k⫻b⫹¹2␮k

T共V_kab^kk ⫺I₃TrV_kab^kk 兲␴^. 共63兲 Finally, it is customary to divide the last contributions into two parts, one of which is traceless. We then obtain

具^a兩共␴"A_k兲共␴"p兲兩b典^⫽i␮k"V_kab^k⫻b⫹²3TrV_kab^bk 共␮k"␴兲

⫺␮k

T共V_kab^bk ⫺¹3I₃TrV_kab^bk 兲␴^, 共64兲

具^a兩共␴"p兲共␴"A_k兲兩b典⫽i␮k"V_kab^a⫻k⫹²3TrV_kab^ak 共␮k"␴兲

⫺␮k

T共V_kab^ak ⫺¹3I₃TrV_kab^ak 兲␴^, 共65兲

具^a兩A_k•p⫹B_k"s兩b典⫽i␮k"V_kab^k⫻b⫺¹3TrV_kab^kk 共␮k"␴兲

⫹¹2␮k

T共V_kab^kk ⫺¹3I₃TrV_kab^kk 兲␴^. 共66兲 In these expressions, the first term is the imaginary singlet PSO contribution, the second term is the real triplet FC con- tribution, and the last term is the real triplet SD contribution.

We have seen that the orbital contributions to PSO, FC, and SD operators may be calculated from the second deriva- tives of the matrix element 具^a兩Vk兩b典. Because of transla-

tional invariance, we need never calculate explicitly more than the derivatives with respect to two of the three centers.

We also note that

V_kab^a⫻b⫽⫺V_kab^a⫻k⫽⫺V_kab^k⫻b, 共67兲 in agreement with the fact that the PSO contribution is the same for all three operators共assuming that the vector poten- tial is divergenceless兲. In the case of a point-charge dipole, the Hermitian operator Eq. 共66兲reduces to the usual expres- sion

具^a兩A_k"p⫹B_k"s兩b典^⫽

冓

^a

^冏

^rkrÃk3p

^冏

^b

冔

^"␮k

⫹8␲

3 具^a兩␦共r_k兲兩b典␮k"s

⫹␮k

T

冓

^a

^冏

^{3rkrkTr⫺k5 rk2I3}

^冏

^b

冔

^{s, 共68兲}

containing the standard PSO, FC, and SD operators for point-charge nuclei.

C. Summary

We have demonstrated that the integrals over the first- order magnetic perturbations may be calculated quite straightforwardly as derivatives of the potential integral. To make clear the close relationship of these integrals to the field and field-gradient integrals, we here list the expressions for the EF, EFG, PSO, FC, and SD integrals:

具^a兩HEF兩b典^⫽Zk“^k具^a兩Vk兩b典^{, 共69兲} 具^a兩HEFG兩b典^⫽⫺Zk“k“k

T具^a兩Vk兩b典^{, 共70兲}

具^a兩H_PSO兩b典⫽i␮k"“kÓb具^a兩Vk兩b典^, 共71兲

具^a兩H_FC兩b典⫽⫺²3␮k"sⵜk

2具^a兩Vk兩b典^, 共72兲

具^a兩H_SD兩b典⫽␮^T共“k“k T⫺¹3I₃ⵜk

2兲具^a兩Vk兩b典^s, 共73兲 where the basic operator Vk is given in Eq.共40兲.

IV. ILLUSTRATIVE CALCULATIONS A. Ground-state hydrogen atom

We have performed calculations on the H atom ground state in basis sets of s-type Gaussians developed by Morgan.¹⁶The exponents␩k

N of an Ns Morgan set are given by

␩k

N⫽ck^⫺␦exp关␣共N^␤⫺k^␤兲兴, k⫽1,2,...,N, 共74兲 with c⫽0.35, ␦⫽0.71, ␣⫽2.74, and ␤⫽¹2(

冑

⁵⫺1). The computed perturbation energies through second order ob- tained with these sets are displayed in Table I. The nucleus has been treated as a point charge and magnetic point dipole.

For accurate computations of the FC term, tight Gaussians—that is, functions with high exponents—are re- quired. Therefore, we have performed a few calculations with Morgan sets to which tight Gaussians have been added.

The exponents of the extra functions have been chosen as ten times the largest exponent of the Morgan set, hundred times

that exponent, and so on. With these tight functions added, the numerical results for E⁰¹and E²¹converge more rapidly to the exact values that with Morgan sets of the same size, as expected.

Moreover, we have computed the FC term E⁰¹of the H atom as a function of the size of the nucleus 共Fig. 1兲. The expectation value of G_␩(r) of the ground-state wave func- tion ␸⁰⁰⫽1/

冑

␲^e⫺r is easily evaluated as

4␲

3 具^G␩共r兲典^⫽4₃

冋 ^冉

^␩2⫹1

^冊

^erfc

冉 ^冑

^1␩

冊

^exp

^冉

^␩1

^冊

^⫺

^冑

^␲2␩

册

^.

共75兲

The Gaussian distribution of the magnetic dipole reduces E⁰¹, but the effect becomes noticeable only for Gaussian exponents␩smaller than about 10⁵. For␩⫽10⁵, the reduc- tion of E⁰¹is almost 1%.

B. H₂^¿molecule ion

Calculations on the H₂^⫹ ground state have been per- formed at an internuclear separation R_H–H⫽2.0 a₀ with a basis set of the form 26s13 p7d3 f 1g. This set has been derived as follows: We have added three diffuse s-type Gaus- sians with exponents 0.007 15, 0.0129, and 0.0232 to the (20⫹3)s Morgan basis共cf. Sec. IV A兲. The 13p, 7d, 3f, and 1g sets have been chosen to contain the exponents 1–13, 4 –10, 6 – 8, and 7, respectively, of the 20s Morgan basis, which has been ordered with increasing exponents from 1 to 20.

Table II shows—in comparison with the results of Ref.

11—the perturbation energies up to third order in the relativ- istic perturbation (E⁶⁰) and up to first order in the magnetic perturbation 共FC term, E⁰¹兲. In our 26s13 p7d3 f 1g basis, the agreement with the results of Ref. 11 is satisfactory. In this basis, we have computed the FC term共on only one of the two nuclei兲as a function of the exponent␩of the Gaussian nuclear model 共Fig. 1兲.

FIG. 1. Hyperfine interaction共E¹⁰in a.u.兲of the H atom共solid line兲and the H2⫹molecule with RH–H⫽2.0 a0共dashed line兲as a function of the exponent

␩of the Gaussian-shaped nucleus. Exact values are plotted for H, while the curve for H₂^⫹has been determined numerically using the 26s13 p7d3 f 1g basis. Asymptotic values for␩→⬁are shown as thin dashed lines.

TABLE I. Perturbation energies共in a.u.兲through sixth order for the H atom ground state. E²⁰and E⁴⁰are relativistic DPT energies, E⁰¹⫽(4␲^/3)具␦^(r)典 is the FC term, and E²¹ is the lowest-order relativistic DPT correction to the FC term.

N^a E⁰⁰ E²⁰ E⁴⁰ E⁰¹ E²¹

10 ⫺0.499 998 68 ⫺0.124 961 85 ⫺0.062 293 51 1.316 526 55 1.883 761 48 20 ⫺0.500 000 00 ⫺0.124 999 91 ⫺0.062 499 26 1.332 528 47 1.992 012 72 30 ⫺0.500 000 00 ⫺0.125 000 00 ⫺0.062 499 99 1.333 266 26 1.999 167 88 40 ⫺0.500 000 00 ⫺0.125 000 00 ⫺0.062 500 00 1.333 325 76 1.999 889 55 50 ⫺0.500 000 00 ⫺0.125 000 00 ⫺0.062 500 00 1.333 332 29 1.999 982 69 9⫹1 ⫺0.499 997 18 ⫺0.124 982 27 ⫺0.062 399 69 1.325 710 24 1.940 217 08 18⫹2 ⫺0.500 000 00 ⫺0.124 999 96 ⫺0.062 499 65 1.333 179 30 1.998 165 70 27⫹3 ⫺0.500 000 00 ⫺0.125 000 00 ⫺0.062 500 00 1.333 327 62 1.999 910 82 36⫹4 ⫺0.500 000 00 ⫺0.125 000 00 ⫺0.062 500 00 1.333 332 96 1.999 992 98 45⫹5 ⫺0.500 000 00 ⫺0.125 000 00 ⫺0.062 500 00 1.333 333 29 1.999 999 16

⬁^b ⫺¹2 ⫺¹8 ⫺16¹

4

3 2

aNumber of s-type Gaussians in the Morgan basis共Ref. 16兲. The K⫹L basis sets have been constructed by augmenting the Ks Morgan sets by L tight functions with exponents 10^M⫻␩1

K( M⫽1,2,...,L).

bExact values.

TABLE II. Perturbation energies 共in a.u.兲 for the H₂^⫹ ground state with RH–H⫽2.0 a0.

Energy This work^a Ref. 11

E⁰⁰ ⫺1.102 634 18 ⫺1.102 634 213

E²⁰ ⫺0.138 332 4 ⫺0.138 332 985

E⁴⁰ ⫺0.041 714 ⫺0.041 727 8

E⁶⁰ ⫺0.028 276 ⫺0.028 32

E⁰¹ 0.961 227

aA basis set of the type 26s13 p7d3 f 1g has been used, cf. Sec. IV B.

C. H₂molecule

The computation of EF’s and EFG’s for Gaussian- shaped nuclei is illustrated with calculations on the H₂mol- ecule at an internuclear distance of 1.4 a₀. The results ob- tained in the standard cc-pVTZ basis are depicted in Fig. 2, showing that only very large nuclear sizes 共i.e., for ␩⬍5兲 will give rise to values for the properties of interest that deviate noticeably from the results obtained with the point nuclear model. For point-shaped nuclei (␩→⬁), E_z

⫽⫺0.5104 a.u. and F_zz⫽⫺0.3802 a.u. at the level of full configuration interaction共CI兲in the cc-pVTZ basis.

We have also computed the diamagnetic spin-orbit 共DSO兲 contribution to the indirect spin-spin coupling con- stant in H₂ at R⫽1.4 a₀ at the Hartree–Fock level in the cc-pVTZ basis共Fig. 3兲. This contribution corresponds to the first term in Eq. 共31兲. Without the projector P, the DSO tensor components amount to J_xx⫽J_{y y}⫽⫺14.894 Hz and J_zz⫽23.793 Hz, yielding a total isotropic DSO contribution of J_DSO⫽¹3Tr(J)⫽⫺1.998 Hz. When we introduce the projector P on an auxiliary small-component basis of the type 2s6 p2d1 f , we obtain J_xx⫽J_{y y}⫽⫺14.969 Hz, J_zz

⫽23.597 Hz, and J_DSO⫽⫺2.114 Hz. The auxiliary basis was obtained by taking the first derivatives with respect to x, y, and z of the primitive functions (5s2 p1d) of the cc-pVTZ basis. This choice corresponds to unrestricted kinetic balance 共UKB兲; see also Ref. 9.

The effect of the projectorPon the DSO tensor compo- nents amounts to a few tenths of a Hz 共ca. 1%兲. Similar effects have been observed in Ref. 9 for the DSO contribu- tions to the spin-spin coupling constants in H₂O, where it was also found that a small-component basis in terms of a restricted kinetic balance共RKB兲appeared to be insufficient.

However, more computations with auxiliary basis sets are required to obtain full insight into the basis-set requirements for the RI approximation in Eq.共31兲.

V. CONCLUSIONS

Matrix elements that occur in a DPT of magnetic prop- erties and relativistic corrections can be computed by pro- jecting the perturbed small components onto an appropriate small-component auxiliary basis set. The relevant matrix el- ements can be simply evaluated for the Gaussian nuclear model, commonly used in relativistic calculations on mol- ecules containing heavy elements. The Gaussian integrals in- volving the PSO, FC, and SD operators are closely related to those of the EF and EFG operators, and may be evaluated as derivatives of the potential integrals.

ACKNOWLEDGMENTS

Much of the material presented here is contained in the Cand. Scient. Thesis of one of the authors共A.C.H.兲, Univer- sity of Oslo, 1999. The authors are aware of similar work carried out by Monika Dahlbeck at the University of Bo- chum. Werner Kutzelnigg and Trond Saue are acknowledged for helpful discussions on the resolution of identity em- ployed in Section II C. Our study has been supported by the Research Council of Norway 共grant of computing time, No.

NN1118K兲. The research of one of the authors 共W.K.兲 has been made possible by a fellowship of the Royal Netherlands Academy of Arts and Sciences.

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FIG. 2. Electronic contribution to the electric field vector component Ez

共dashed line兲and electric field-gradient tensor component F_zz共solid line兲at one of the two nuclei of the H2molecule共RH–H⫽1.4 a0兲as a function of the exponent␩of the Gaussian-shaped nucleus, as obtained from full CI calcu- lations with the cc-pVTZ basis set. Asymptotic values for␩→⬁are shown as thin dashed lines.

FIG. 3. DSO contributions 共dashed line: Jxx⫽Jy y, solid line: Jzz兲to the indirect spin–spin coupling constant of the H₂molecule共R_H–H⫽1.4 a₀兲as a function of the exponent ␩ of the Gaussian-shaped nucleus, as obtained from Hartree–Fock calculations with the cc-pVTZ large-component and corresponding UKB 2s6 p2d1 f small-component basis sets. Asymptotic values for␩→⬁are shown as thin dashed lines.

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