Chemical Physics 401 (2012) 103–112

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Chemical Physics

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Nuclear size effects in rotational spectra: A tale with a twist

Stefan Knecht a, Trond Saue b,⇑a Department of Physics and Chemistry, University of Southern Denmark, Campusvej 55, 5230 Odense M, Denmarkb Laboratoire de Physique Quantique (CNRS UMR 5626), IRSAMC, Université Paul Sabatier, 118 Route de Narbonne, F-31062 Toulouse cedex, France

a r t i c l e i n f o

Article history:Available online 6 November 2011

Keywords:Relativistic quantum chemistryNuclear size effectsDiatomic moleculesContact electron densityMolecular properties

0301-0104/$ - see front matter � 2011 Elsevier B.V. Adoi:10.1016/j.chemphys.2011.10.030

⇑ Corresponding author.E-mail addresses: knecht@ifk.sdu.dk (S. Knecht),

(T. Saue).

a b s t r a c t

We report a 4-component relativistic benchmark study of the isotopic field shift in the rotational spec-trum of three diatomic molecules: TlI, PbTe and PtSi. A central quantity in the theory is the derivativewith respect to internuclear distance of an effective electron density associated with a given nucleus, cal-culated at the equilibrium distance. The effective density, which is related to the mean electron densitywithin the nuclear volume, is usually replaced with the contact density, that is, the electron density at theorigin of the nucleus. Our computational study shows that for the chosen systems this induces errors onthe order of 10%, which is not acceptable for high-precision work. On the other hand, the systematic nat-ure of the error suggests that it can be handled by an atom-specific correction factor. Our calibrationstudy reveals that relativistic effects increase the contact density gradient by about an order of magni-tude, and that the proper transformation of the associated property operator is mandatory in 1- and 2-component relativistic calculations. Our results show very good agreement with the experimental datapresented by Schlembach and Tiemann [Chem. Phys. 68 (1982) 21], but disagree completely with therevised results given by the same group in a later paper [Chem. Phys. 93 (1985) 349]. We have carefullyre-derived the relevant formulas and cannot see that the rescaling of results is justified. Curiously previ-ous DFT calculations agree quite well with the revised results for TlI and PbTe, but we demonstrate thatthis is because the authors inadvertently employed a non-relativistic Hamiltonian, which by chanceinduces an error of the same magnitude as the suggested scaling. For the PtSi molecule our results forthe correction term due to nuclear volume disagree with experiment by a factor five, and we recommenda re-examination of the experimental data.

� 2011 Elsevier B.V. All rights reserved.

1. Introduction

The incompatibility of the a-scattering experiments of Geigerand Marsden [1] with the multiple scattering predicted by Thom-son’s ‘‘plum pudding’’ atomic model [2] led Rutherford in 1911to propose that all positive charge was concentrated in a nucleusat the center of a homogeneously negatively charged sphere[3,4]. Rutherford concluded that the size of the nucleus had to bevery much smaller than the atomic radius, on the order of10�15 m or less. Since the nuclear size is indeed negligible for thedescription of most phenomena on an atomic scale, an overwhelm-ing majority of theoretical studies of molecules and solids, startingwith the celebrated scattering formula of Rutherford, treat nucleias point charges. A notable exception are relativistic molecularelectronic structure calculations based on the finite basis approxi-mation, where the point nucleus assumption leads to basis set con-vergence problems due to the weak singularities displayed by thesolutions of the Dirac Hamiltonian at the nuclei. To overcome this

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trond.saue@irsamc.ups-tlse.fr

problem it was early realized [5,6] that the introduction of a finite(Gaussian) nuclear charge distribution not only curbs the singular-ity but also leads to Gaussian type solutions near the origin, whichgreatly alleviates the integral evaluation in relativistic wave func-tion calculations using basis set expansions.

However, many phenomena known from atomic and molecularspectroscopy as well as isotope chemistry even lack a reasonableexplanation if the finite size of a nucleus is not taken into account.A notable example is the isotope shift in the electronic spectra ofatoms and molecules [7–9] which is well understood as an inter-play between (i) the mass shift, scaling approximately with DM/M2 (M being the mass), and (ii) the nuclear field shift, arising fromthe difference in size and shape of the extended nucleus of eachisotope. The mass shift term typically becomes less significant withincreasing nuclear charge values Z due to its scaling propertieswhereas the nuclear field shift is known to be the main contributorto the total isotope shift for Z P 58 [8]. Following similar lines ofthoughts Bigeleisen [10,11] developed in the mid 1990s a generaltheoretical model which was able to explain isotopic anomaliesin chemical reactions, as for example observed in U(IV)–U(VI) ex-change reaction experiments [12] which yielded a 235U isotopicenrichment over 238U in the U(VI) species of an initial 235U/238U

104 S. Knecht, T. Saue / Chemical Physics 401 (2012) 103–112

mixture. As a result, Bigeleisen could unambiguously attribute theexperimentally observed excess separation factor (beyond the ex-pected regular mass-dependent fractionation) to a nuclear fieldshift effect rather than a nuclear spin effect [12]. The theory provedalso successful to the description of related findings in 233U/238Uisotope studies [13] and ab initio atomic and molecular studies ofuranium isotope fractionation have been reported by Abe andco-workers [14,15]. The effect of nuclear volume has also been con-sidered as a possible source of explanation for isotopic anomaliesobserved in the isotope fractionation of other heavy atoms suchas mercury [16–19], as well as observed in the early solar system[20].

In the present work we consider the effect of nuclear volume onrotational spectra. In the early 1980s Tiemann and co-workers [21]conducted a series of high-resolution rotation spectroscopy exper-iments for isovalent, closed-shell diatomics AB, which aimed at anaccurate assessment of spectroscopic constants as for example, theequilibrium bond length Re or the rotational constant Be. In the par-ticular case of thallium and lead compounds, they were not able tobring their measured isotope data in agreement using the existingtheory of adiabatic and non-adiabatic correction terms to theBorn–Oppenheimer (‘‘clamped nuclei’’) approximation [22–24],considering their order of magnitude required to derive consistentDunham coefficients Ykl [25]. In a follow-up publication Knöckeland Tiemann [26] therefore first unambiguously identified thenecessity of, beyond the known mass-dependent corrections,introducing an additional correction factor arising from the finitesize of the nuclei. Taking into consideration this finite extensionin the initial molecular Hamiltonian, Schlembach and Tiemann[27] built a sound theoretical foundation of the molecular fieldshift in rotational spectra. Their final expression for the total cor-rection term to the first Dunham coefficient Y01, which is of parti-cular interest in rotational spectroscopy, thus not only containsmass-dependent contributions but also a further term that is pro-portional to the change in the mean-square nuclear charge radiusdhr2

ni times the derivative dqð0ÞdR

� �jRe

of the electron density q(0) ata given nucleus with respect to the internuclear distance R, takenat Re. The so-called contact density q(0) is a central quantity inthe theory of the Mössbauer isomer shift [28–33], but appears asa consequence of approximations [34,35]. The same holds true inthe case of the molecular field shift, as will be discussed in Section2.

A careful review of the original spectroscopic data for the hea-vy-element Pb- and Tl-compounds [21] by means of the expres-sions derived in Ref. [27] revealed then the molecular field shiftterm as main contributor to the isotope shift corrections to Y01.Interestingly, though, in 1985 [36] the same experimental groupannounced a revision of their entire set of earlier results (for a re-view see for example Ref. [37]) based on a computational errorwhere they recommend a scaling of the data by a factor 10. Theirrevised findings have later been corroborated by Cooke and co-workers [38–41] who performed a series of (scalar-relativistic) abinitio calculations based on density-functional theory (DFT). Theyfurthermore demonstrated the existence of a molecular field shifteffect in the rotational spectrum of platinum silicide [39] by carry-ing out high-resolution experiments accompanied by DFTcalculations.

The purpose of the present study is to provide an independentcheck on both theoretical and experimental studies of the isotopicfield shift in rotational spectra. On the theoretical side it is knownthat 1- and 2-component relativistic calculations of molecularproperties probing the electron density near nuclei are highly sen-sitive to picture change errors [42–45] and so we wanted to cali-brate previous calculations against 4-component relativistichighly correlated calculations. Theoretical studies can also shedlight on the major physical contributions to the molecular field

shift. The paper is therefore organized as follows: In Section 2 wecarefully re-derive an expression for the first-order modificationof the rotational constant Be due to nuclear volume changesbetween isotopes. We have next carried out non-relativistic,scalar-relativistic 2-component as well as relativistic two- and4-component ab initio DFT and high-level wave function CoupledCluster calculations for three representative molecules, namelyTlI, PbTe and PtSi. Computational details are provided in Section3, and in Section 4 we compare our results to previous theoreticaland experimental results based on the original as well as revisedexpressions derived by Tiemann and co-workers. By means ofour extensive reference data we shall reveal a curious twist inthe tale before concluding in Section 5.

2. Theory

Within the Born–Oppenheimer (‘‘clamped nuclei’’) approxima-tion the rovibrational energy levels of a closed-shell diatomic mol-ecule AB may be determined from an effective radial Schrödingerequation of the form

� �h2

2ld

dR2 þ EelðRÞ þ �h2JðJ þ 1Þ2lR2

" #wm;JðRÞ ¼ Em;Jwm;JðRÞ ð1Þ

where R is the internuclear coordinate and l the reduced mass.Solutions to this problem of rotating vibrator were provided byDunham [25] in the framework of the Jeffreys–Wentzel–Kramers–Brillouin (JWKB) approximation [46–49,25] and expressed in theform nowadays known as the Dunham expansion

Em;J ¼ hXk¼0

Xl¼0

Yklðmþ 1=2Þk½JðJ þ 1Þ�l ð2Þ

where m and J are vibrational and rotational quantum numbers,respectively. Eel is the potential obtained from solving the electronicproblem

Helwelðrel; RÞ ¼ EelðRÞwelðrel; RÞ ð3Þ

where rel designates all electronic coordinates. The electronicHamiltonian, relativistic or not, has the generic form

Hel ¼X

i

hðiÞ þ 12

Xi–j

gði; jÞ þ VAB; h ¼ h0 þ VeN ð4Þ

where VAB is the classical repulsion of nuclei A and B and g(i, j) thetwo-electron operator. The one-electron operator h splits into thefree-particle Hamiltonian h0 and a term VeN describing the interac-tion with the nuclei.

Following Schlembach and Tiemann [27] we give the Dunhamcoefficients Ykl in units of frequency. As emphasized by Ogilvie[50,51] they are not freely adjustable fitting coefficients since theyare interrelated. To lowest order in m and J one has [25]

Y10 ¼xe

2p1þ B2

e=4m2e

� �½. . .�

h i� me

Y01 ¼ Be 1þ B2e=2m2

e

� �½. . .�

h i� Be ¼

h

8plR2e

ð5Þ

where Re is the equilibrium bond distance. For a specific isotopomera of the molecule AB the Dunham coefficients are given by

Yakl ¼ l�ðk=2þlÞ

a Ukl ð6Þ

where Ukl are isotope-independent coefficients. However, the aboverelation supposes (i) that all isotopomers experience the sameinternuclear potential and (ii) that the first-order JWKB approxima-tion (the Bohr–Sommerfeld quantization condition) is exact [52]. A

S. Knecht, T. Saue / Chemical Physics 401 (2012) 103–112 105

more general expression, taking into account the breakdown of theBorn–Oppenheimer approximation, was proposed by Ross et al. [53]in an experimental study of isotopomers of CO

Yakl ¼ l�ðk=2þlÞ

a Ukl 1þ me

MADA

kl þme

MBDB

kl

� �ð7Þ

and later developed theoretically by Bunker [23] and Watson [24].In the above expression DA

kl and DBkl are mass-independent atom-

specific correction factors.In the following we shall focus on the modification of the rota-

tional constant by modification of the size of one nucleus, say nu-cleus A. Experimentally only discrete values of the nuclear radiusare available through isotopic substitution A ? A0, but for the pur-pose of derivation it is more useful to consider the electronic en-ergy Eel as a continuous function of both internuclear distance Rand the nucleus radius n, that is Eel � Eel(R, n). We shall let nA cor-respond to the nuclear radius of a particular reference isotope ofatom A. The equilibrium internuclear distance Re for any value ofnuclear radius n is found by minimizing the electronic energy withrespect to internuclear distance,

dEel

dR

�����n

¼ 0; ð8Þ

and thus becomes a function of the nuclear radius parameter, whichallows us to attack the above problem by variational perturbationtheory (see for instance Ref. [54]). Since the variational conditionEq. (8) is valid for any nuclear radius n we find the first-order shiftin the equilibrium distance by derivation with respect to nuclear ra-dius n

d2Eel

dndR

�����nA

¼ d2Eel

dR2

dRdnþ @

@ndEel

dR

!" #nA

¼ 0 ð9Þ

The expression for the first-order equilibrium distance, Eq. (9),contains the explicit derivative of the electronic energy Eel with re-spect to nuclear radius. Such derivatives also appear in recenttheoretical studies of the isomer shift in Mössbauer spectroscopy[55,56,35]. We can safely ignore contributions from the classicalrepulsion VAB of nuclei such that the only non-zero contributioncomes from the electrostatic interaction between the electronsand nucleus A. Its modification upon the isotope substitutionA ? A0 is given by

dEelA0A ¼

Zqeðre; RÞ½/Aðre; n

A0 Þ � /Aðre; nAÞ�dse ð10Þ

It should be noted that in the above expression we ignore the impli-cit dependence on nuclear radius of the electronic charge distribu-tion qe. We express the scalar potential of nucleus A in terms of thenormalized nuclear charge distribution qA

/Aðre; nÞ ¼Ze

4pe0

ZqAðrn; nÞ

rnedsn ð11Þ

The perhaps simplest model for a finite nuclear charge distributionis the homogeneous charged sphere of radius n

qHn ðrÞ ¼

q0; r 6 n

0; r > n

�; q�1

0 ¼4p3

n3 ð12Þ

which is also the model considered by Schlembach and Tiemann[27]. The associated potential is

/Hn ðrÞ ¼

Ze8pe0n 3� r2

n2

� �; r 6 n

Ze4pe0r ; r > n

8<: ð13Þ

More widely employed in relativistic molecular calculations is theGaussian model [5,6] in which the charge distribution is given by

qGn ðrÞ ¼ q0 exp �r2=n2

G

� ; q�3

0 ¼ p1=2nG ð14Þ

with the associated potential

/GnðrÞ ¼

Ze4pe0r

erfðr=nGÞ: ð15Þ

The radius parameter nG can be connected to the radius n of thehomogeneous sphere by requiring identical second radial momenta

hr2ni ¼

Zr2

nqnðrnÞdsn ¼32

n2G ¼

35

n2; ) nG ¼ffiffiffi25

rn ð16Þ

Schlembach and Tiemann [27] consider the modification of theinternuclear potential when going from a point nucleus to a finitenucleus for each isotope, that is

dEelA0 ¼

Zqeðre; RÞ½/Aðre; n

AÞ � /Aðre; 0Þ�dse ð17Þ

in order to introduce mass-independent Dunham coefficients Ukl fora fictitious molecule of point-like nuclei. However, such an ap-proach is problematic in the relativistic case since the electronicdensity displays a weak singularity at point nuclei. As such, it is per-haps better to modify the Dunham expansion by the introduction ofa reference isotopomer, as suggested by Le Roy [52]. However, wecan formally write Eq. (10) as

dEelA0A ¼ dEel

A00 � dEelA0 ð18Þ

where the electron density will be that of a molecule with finite sizenuclei. In passing we note that the difference potential in Eq. (17),using the homogeneous charged sphere model, looks curiously sim-ilar to the potential of the Rutherford atom, consisting of a positivepoint charge at the origin and the homogeneous electronic chargewithin a sphere of radius n [3]. We can formally write Eq. (17) as

dEelA0 ¼ �qe

Z½/Aðre; n

AÞ � /Aðre; 0Þ�dse ð19Þ

where we have introduced a constant effective electronic chargedensity �qe. Due to the extreme short-range nature of the above dif-ference potential, as seen for instance from Eq. (13), the effectivedensity �qe is typically approximated by the contact density, thatis the value of the electronic charge density at the nucleus:

�qe � qeð0Þ ð20Þ

For light atoms this is certainly an excellent approximation; for themore extended nuclei of heavier atoms this leads to an overestima-tion [34]. In the case of the Mössbauer isomer shift the deviation isquite systematic in nature, which suggests that it can be handled bya correction factor [35], a feature that will also be investigated inthe present study.

Using Eq. (18) we obtain the expression

dEelA0A ¼ �qe

Ze6e0

dhr2niA0A; dhr2

niA0A ¼ hr2niA0 � hr2

niA ð21Þ

which holds for both the homogeneous charged sphere and theGaussian model of the nuclear charge distribution. FollowingFilatov [55] an alternative approach to dEel

A0A is to approximate thepotential difference of Eq. (10) by a first-order Taylor expansion,that is

dEelA0A �

Zqeðre; RÞ@/Aðre; nÞ

@n

����nAðnA0 � nAÞdse ¼

@

@ndEel

dR

!�����nA

ðnA0 � nAÞ

ð22Þ

thus connecting to the expression for the first-order equilibriumdistance, Eq. (9).

106 S. Knecht, T. Saue / Chemical Physics 401 (2012) 103–112

Using this connection we find that the change in equilibriumbond distance due to change in nuclear size upon isotope substitu-tion A ? A0 can be expressed as

dRe ¼ �Ze2

6e0kAe

dhr2niA0A

d�qe

dR

� �RA

e

ð23Þ

where RAe and kA

e corresponds to the equilibrium bond distance andthe force constant of the reference isotopomer. The correspondingmodification of the equilibrium rotational constant Be is

dBe ¼ BAe�2Re

� �dRe ¼ BA

e VAdhr2niA0A ð24Þ

where appears the factor

VA ¼ Ze2

3e0kAe RA

e

d�qe

dR

� �RA

e

ð25Þ

It has exactly the same form as Eq. (25) of the 1982 paper bySchlembach and Tiemann [27] except that the spectroscopic con-stants RA

e and kAe are that of the reference isotopomer and not of a

fictitious molecule with point-like nuclei. Curiously, in a paper[36] from 1985 by the same group the formula for the isotopic fieldshift factors is given with an additional factor p2

VA ! p2VA ð26Þ

which to our opinion is not justified. However, as we shall see inSection 4 it contributes to a strange twist in our story.

3. Computational details

All molecular calculations reported in this paper have been car-ried out with the DIRAC10 program package [57].

3.1. Electron correlation methods

The absolute magnitude of the electron density in the vicinity ofa given nucleus is well described within a mean-field approach tothe electron–electron interaction since the dominant part is gov-erned by the influence of the nuclear potential and deformationsof core orbitals are expensive in energy. The significant relativechanges in the contact density, which yield the most sensitive con-tributions to the isotopic field shift, are, on the other hand, affectedby subtle alterations in the valence electronic structure in a vary-ing chemical environment. An accurate description of these va-lence contributions thus requires proper inclusion of electron–electron correlation, as has been shown recently in the context ofisomer shift predictions for Mössbauer spectroscopy [58,35]. Forthe three closed-shell diatomics TlI, PbTe and PtSi we thereforeperformed single-reference coupled-cluster (CC) calculations witha full iterative treatment of single and double excitations (CCSD)and including perturbative corrections for triple excitations(CCSD(T)) [59–61].

The central object of our study is the first geometrical derivativeof the contact density for selected nuclei X calculated at the equi-librium internuclear distance Re

q½1�X0 ¼ dqXð0ÞdR

jReð27Þ

which we in the following shall refer to as the contact density gra-dient. Presently there is no analytic implementation of CC expecta-tion values in the DIRAC10 program package. For the calculation ofcontact densities at the CC level we therefore pursued a finite-field(ff) approach using a computational protocol which is described atlength elsewhere [35] and shall only be briefly sketched in thefollowing. Exploiting the additivity of the contact density

contributions q(0) = qHF(0) + qcorr(0), namely, (i) the Hartree–Fockexpectation value of qHF(0), which can be evaluated analytically inDIRAC10, and (ii) the electron–electron correlation term qcorr(0),we are only left with the determination of the latter contributionin a finite-field scheme. In line with our previous ff-coupled clusterproperty calculations [35] we employed an optimal finite-fieldparameter of 10�8 for all diatomic systems under considerationand took take advantage of the central-difference method [62]using a seven-point stencil for the numerical differentiation. In or-der to obtain the contact density gradient q½1�X0 we performed a sec-ond numerical differentiation by means of the central-differencemethod using a step size of 0.0125 Å as ‘‘finite-field’’ parameter.Computing the contact density gradient as derivative of a polyno-mial data fit function yielded equally identical results.

Besides the isotopic field shift evaluations based on the wavefunction Hartree–Fock and CC methods we also carried out 4-and 2-component density functional theory (DFT) calculations.Aiming at an assessment of both the accuracy and internal consis-tency within the DFT contact densities (and therefore the isotopicfield shift) we employed an ample set of exchange-correlationfunctionals, namely LDA (VWN5) [63,64], BLYP [65–67], B3LYP[65,68,69], CAMB3LYP [70], PBE [71], PBE0 [72], and furthermorethe SAOP model potential [73]. Of main concern for electron den-sity evaluations in the core region is the use of a sufficiently denseintegration grid in the numerical integration of the exchange-cor-relation evaluation [74]. We met this particular requirement byemploying throughout all DFT calculations an ultrafine grid.

3.2. Hamiltonian

An important aspect of the present study is furthermore to ad-dress successive approximations to the 4-component DC Hamilto-nian and their validity in the evaluation of the field shift effect for agiven nucleus. In particular, we therefore compare 4-componentrelativistic results to:

� Relativistic and scalar-relativistic data using the eXact2-Component (X2C) Hamiltonian [75].� 4-Component spin–orbit free (sf) [76,77] (scalar-relativistic)

results.� Non-relativistic (NR) values employing the Lévy–Leblond [78]

Hamiltonian.

The X2C calculations have been carried out either including 2-electron spin-same-orbit corrections provided by the AMFI[79,80] code (relativistic) or by retaining only the spin-free termsin the one-electron Hamiltonian prior to the transformation to 2-component basis (scalar-relativistic; in the following denotedX2C-sf).

For reasons of computational efficiency, a molecular mean-fieldapproximation 4cDC⁄⁄ to the 4-component DC Hamiltonian 4cDCwas applied in the majority of CCSD(T) calculations, where ournotation strictly follows the Hamiltonian hierarchy introduced inRef. [81]. The relative deviation in the contact density gradientcompared to the exact 4cDC-CCSD(T) value was in all cases testedless than 0.05%.

3.3. Basis sets

All molecular calculations were performed using atom-centeredlarge-component basis sets consisting of uncontracted scalarGaussian type orbitals (GTO) where the small-component basisfunctions were generated, where appropriate, by the restrictedkinetic balance condition as implemented in DIRAC10 [82]. Inthe case of the heavy elements Pb, Te, Tl, I and Pt we used the

S. Knecht, T. Saue / Chemical Physics 401 (2012) 103–112 107

triple-f (TZ) and quadruple-f (QZ) basis sets of Dyall [83–85]. Thebasic large-component SCF set of primitives was further aug-mented with the recommended correlating and polarizing func-tions in order to properly account for correlation contributionsfrom the nsp valence and outer-core (n � 1)d shells. For the Pt TZ

basis we also added a set of 2g1h primitives to allow for correlationof the (n � 2)f electrons.

Basis set saturation at the heavy nuclei of interest with respect toan accurate computation of the contact density was achieved by fur-ther augmenting the set of primitives in an even-tempered fashionwith two more tight s and one tight p functions, which is in line withprevious studies [86,35]. The final large-component basis sets(denoted TZ + 2s1p (QZ + 2s1p) in the following) thus read as[33s28p18d12f1g] ([37s33p22d18f4g1h]) for Pb and Tl, [29s22p16d4f1g] ([34s28p19d6f4g1h]) for Te and I, and [32s25p15d11f4g1h]for Pt, respectively. As Si basis set the correlation-consistent Dun-ning basis set [87] of triple-f quality (ATZ) was likewise chosen infully uncontracted form and augmented with diffuse functions.

3.4. Active space considerations

We have shown in a recent benchmark work for mercury-con-taining compounds [35] that an accurate assessment of correlationeffects to the contact density q(0) by means of wave-functionbased methods necessitates the inclusion of both core-valenceand valence correlation contributions. Our active space in all fi-nite-field coupled-cluster calculations therefore comprised the5d6s6p shells of Tl (Pb) and 4d5s5p shells of I (Te) of the thalliumiodide (lead telluride) compound. For the platinum silicide diatomthe active space has been adapted accordingly with an explicit cor-relation treatment of the Pt 4f5p5d6s6p and Si 2s2p shells.

The size of the virtual spinor space for all three molecular sys-tems was tailored to contain all recommended core- and valencecorrelation as well as valence dipole polarization functions. Thiscorresponds for the TlI and PbTe compounds to a threshold of40 hartree whereas for the PtSi diatom the cutoff is fixed at 62 har-tree. The validity of this choice was confirmed by calculating thecontact density gradient q½1�Tl

0 in thallium iodide at the CCSD(T) le-vel using an enlarged virtual space threshold of 134 hartree. Grad-ually saturating the unoccupied space by this means led to a (forthe present purpose) negligible decrease of 0.1% in the final valueof q½1�Tl

0 .

Table 2Atomic matrix elements for the Thallium atom (HF/TZ + 2s1p), comparing the contactqe(0) and effective �qe (number) density. All values are in atomic units a�3

0 .

qe(0) �qe � qeð0Þ

1s1/2 2112536.88 �215363.37

4. Results and discussion

4.1. Molecular structures

Table 1 compiles the equilibrium bond distances Rcompe and

vibrational frequencies xcompe that have been derived from our

CCSD(T)/TZ + 2s1p data as a by-product of the contact density cal-culations. Considering first the equilibrium bond distance Re, we

Table 1Spectroscopic constants for the 1Rþ0 ground state of 205Tl127I, 208Pb130Te, and 195Pt28Sicomputed at the molecular-mean-field 4DC⁄⁄-CCSD(T)/TZ + 2s1p level of theory (thereader may refer to the text for more details on the computational setup). Note thatour values are not corrected for basis-set superposition errors.

Molecule Rcompe [Å] Rexp

e [Å] xcompe [cm�1] xexp

e [cm�1]

TlI 2.838 2.8136a 147.5 �150a

PbTe 2.612 2.5949a 211.0 211.9a

PtSi 2.083 2.0615b (2.0629)c 544.9 549d

a Ref. [93].b Ref. [39].c Ref. [94].d Ref. [95].

find a very good agreement with the experimental data, with ourvalues consistently being only 0.02 Å longer than their reference.Since our primary concern in the present study was not to repro-duce experimental spectroscopic constants to highest precisionwe did not further pursue a basis set superposition error correc-tion. Turning to the vibrational frequencies xe, we observe a sim-ilar good performance of our CCSD(T)/TZ + 2s1p data with thelargest discrepancy being less than 2% for thallium iodide. More-over, the present 4-component molecular mean-field CCSD(T) val-ues for platinum silicide evidently improve upon earlier theoreticalestimates based on a CASPT2 study by Barysz and Pyykkö[88] whoreported spectroscopic constants of Re = 2.1 Å and xe = 531 cm�1

for this molecule.We conclude this paragraph by noting that all geometrical

derivatives, subject to discussion as follows, have been taken atthe experimental equilibrium internuclear distances Rexp

e summa-rized in Table 1. This practice ensures a fair comparison with thecomputational results by Cooke et al. [38,39] that is to be discussedin Section 4.5.

4.2. On the use of the contact density

Before embarking on the analysis and interpretation of our geo-metrical derivative data of the electronic charge density in the lightof experimental and earlier theoretical predictions, we first inves-tigate the validity of our approximation (20) of the effective den-sity �qe by the contact density qe(0). This approach leads to amodified expression of Eq. (25), which describes the factor VA con-nected to the molecular field shift:

VA � Ze2

3e0kAe RA

e

q½1�A0 ð28Þ

This is the formula from which we have derived our VA values thatenter the discussion in Section 4.5.

Table 2 compiles for the Thallium atom individual orbital con-tributions to the effective density relative to the contact density,which were computed at the HF/TZ + 2s1p level. The evaluationof effective densities has recently been made available in a devel-opment version of DIRAC10 and its implementation is described infull detail in Ref. [35]. Two trends are clearly discernible from Table2: (i) the contact density is solely composed of contributions froms1/2 and p1/2 shells (as expected from theory) whereas the effectivedensity yields additional contributions considerable in their mag-nitude first and foremost from p3/2 orbitals. The latter findings

2s1/2 322388.19 �33035.873s1/2 74350.56 �7625.344s1/2 18799.34 �1928.445s1/2 3695.82 �379.146s1/2 371.46 �38.112p1/2 24695.53 �2449.092p3/2 0 2 � 0.563p1/2 6395.44 �634.803p3/2 0 2 � 0.164p1/2 1598.33 �158.674p3/2 0 2 � 0.085p1/2 280.27 �27.825p3/2 0 2 � 0.016p1/2 2.46 �0.246p3/2 0 < 0.01

Total 2565114.28 �261639.27

108 S. Knecht, T. Saue / Chemical Physics 401 (2012) 103–112

can be understood as a result of sizable p3/2 orbital values withinthe nuclear volume. (ii) Though contributions from the s1/2 andp1/2 orbitals are throughout significantly higher in the contact den-sity than in the effective density approach, they appear to be on arather systematic basis of �+10%, a fact, which has also been ob-served for the mercury atom [35]. We expect the contact densityto be an increasingly better approximation of the effective densitywith decreasing nuclear charge; for xenon the deviation is found tobe around 5% [89].

Significant for the present discussion of the molecular field shiftis, however, the first geometrical derivative rather than the absolutemagnitude of the electronic charge density at a given nucleus andinternuclear distance. Table 3 summarizes for our three referencemolecules the effective as well as contact density gradients q[1]X

computed at the 4-component HF/TZ + 2s1p and DFT/TZ + 2s1p

level, respectively. Yet again we find considerable deviations forthe first derivative between both approaches. The use of the contactdensity approximation for the study of nuclear size effects in therotational spectra of Pt, Tl, and Pb diatomics, may, nevertheless,be well justified since the deviations exhibit a systematic natureon the order of 10% and, equally important, are independent ofthe level of electron–electron correlation included.

4.3. Projection analysis of the contact density

At the SCF level the electronic density at nuclear center X maybe written as

qX ¼ �eXnocc

i

wijdðr� RXÞjwih i ð29Þ

Inserting a linear combination of atomic orbitals (LCAO) expansionof the molecular orbitals into the above expression leads to a pro-jection analysis [90,35] of expectation values particularly usefulfor local properties such as the contact density. In practice theLCAO-expansion is limited to atomic orbitals p which are occupiedin the ground states of atoms P

jwii ¼X

pP

wPp

��� EcP

pi þ wpoli

��� Eð30Þ

and to which is added the orthogonal complement wpoli which is

denoted the polarization contribution. Expectation values areaccordingly decomposed into inter- and intra-atomic as well aspolarization contributions. As shown in Ref. [35] the contact densityqX

0 is completely dominated by the intra-atomic contribution ofsame center X and can accordingly be expressed as

qX0 � �e

Xpq

RLpRL

q þ RSpRS

q

n or¼RX

DXXqp ; DQP

qp ¼X

i

cQqic

P�pi ð31Þ

Table 3Effective and contact electron density gradients, respectively, at the nuclei Tl, Pb, andPt calculated at the 4-component Dirac-Coulomb level using the TZ + 2s1p basis. Thederivatives (in �4) are taken at the respective experimental geometries of TlI, PbTe,and PtSi.

Method q½1�X0�q½1�X Dð�q½1�X � q½1�X0 Þ

TlIHF 212.15 190.29 �10.3%DFT/PBE 114.53 102.67 �10.4%DFT/SAOP 119.96 107.60 �10.3%

PbTeHF 240.18 215.00 �10.5%DFT/PBE 155.74 139.29 �10.6%

PtSiHF �1183.83 �1070.52 �9.6%DFT/PBE �656.39 �593.34 �9.6%

where RL and RS are the large and small component radial functions,respectively. Non-zero contributions to the contact density are pro-vided exclusively by the large components of s1/2 orbitals and thesmall components of p1/2 orbitals, whereas for the effective density�qe other orbitals may come into play, as seen in Section 4.2. The ra-dial functions RX are evidently independent of internuclear distance,so all geometry-dependence arises from the density matrix DXX

qp ex-pressed in terms of atomic orbital expansion coefficients.

From Table 4 it can be seen that the intra-atomic contributionfrom the Thallium atom dominates the contact number densitygradient q½1�Tl

0 in TlI. Assuming generality of this result, we canaccordingly express the contact density gradient as

q½1�X0 � �eX

pq

RLpRL

q þ RSpRS

q

n or¼RX

dDXXqp

dR

!�����Re

: ð32Þ

It is furthermore seen that a negative contribution from diagonalelements (p = q), which would contribute to the atomic expectationvalue, are overwhelmed by positive off-diagonal contributions,which come into play due to the breakdown of atomic symmetryin the molecule. The two opposing contributions reflect re-organization of the electron density as a function of internucleardistance. A more detailed breakdown of the expectation value isseen in Table 5. Looking first at just the geometric D[1]Tl derivativeof the atomic density matrix, it can be seen that the above-mentioned density re-organization takes predominantly place, asexpected, amongst the valence orbitals. However, due to the verymuch larger values of the radial functions of core orbitals at thenucleus, the contribution of core orbitals to the overall expectationvalue is not negligible. Finally, it may be noted that there are non-zero contributions from the p1/2 orbitals, but significantly smallerthan the contributions from s1/2 orbitals.

4.4. Effect of Hamiltonian, basis and method

We summarize the results of our calculations on TlI, PbTe andPtSi in Tables 6–8, respectively. Before comparing with availableexperimental data we consider the effect of Hamiltonian, methodand basis sets on the calculated geometrical density derivatives.Starting from the relativistic DFT(SAOP) value q½1�Tl

0 ¼ 119:96 Å � 4

based on the 4-component relativistic Dirac-Coulomb (4DC)Hamiltonian obtained for TlI at the TZ + 2s1p basis level, we seethat going to the non-relativistic (NR) limit gives a reduction of�82%, or one order of magnitude, showing the importance of rela-tivistic effects. Treating in turn exclusively scalar-relativistic ef-fects at the spin–orbit free level (sf), results in a slight overshootby +12% for q½1�Tl

0 at the DFT(SAOP)/TZ + 2s1p level, indicating thenecessity to account for spin–orbit coupling contributions for thisproperty irrespective of the closed-shell character of the molecularspecies.

Consider next the ability of the eXact 2-Component (X2C)Hamiltonian to reproduce the 4DC results. Of particular concernin 2-component relativistic calculations are picture change errors[42–44]. The 2-component relativistic one-electron Hamiltonian

Table 4Projection analysis of the contact density gradient q½1�Tl

0 (in �4) in TlI calculated at the4-component DFT(PBE0) level using the TZ + 2s1p basis.

Intra-atomic contribution Tl Total 103.47

Diagonal (p = q) �135.91Hybridization (p – q) 243.99

I Total 0.00Inter-atomic contribution 0.30Polarization contribution 32.63

Table 5Detailed analysis of individual orbital contributions to geometric derivative of atomicdensity matrix D½1�Tl

qp and contact density gradient q½1�Tl0 (in Å�4) in TlI calculated at the

4-component DFT(PBE0) level using the TZ + 2s1p basis. For each quantity contribu-tions from s1/2 and p1/2 orbitals are given in the lower and upper triangle of the table,respectively.

Table 6Contact density gradient q½1�Tl

0 (in �4) of the TlI molecule computed at various level oftheory. The derivative is calculated at the experimental equilibrium interatomicdistance Rexp

e ¼ 2:8136 Å [93].

Method qe Evaluation Hamiltonian Basis set q½1�Tl0

TlIHF w�w 4cDC TZ 209.30HF w�w 4cDC TZ + 2s1p 212.15HF w�w 4cDC QZ + 2s1p 212.07DFT/SAOP w�w NR TZ + 2s1p 21.09DFT/SAOP w�w X2C TZ + 2s1p 461.57DFT/SAOP hqTli X2C TZ + 2s1p 120.50DFT/SAOP w�w 4cDC-sf TZ + 2s1p 134.48DFT/SAOP w�w 4cDC TZ + 2s1p 119.96DFT/SAOP w�w 4cDC QZ + 2s1p 119.75DFT/LDA w�w 4cDC TZ + 2s1p 106.64DFT/PBE w�w 4cDC TZ + 2s1p 114.53DFT/BLYP w�w 4cDC TZ + 2s1p 107.09DFT/PBE0 w�w 4cDC TZ + 2s1p 140.63DFT/B3LYP w�w 4cDC TZ + 2s1p 127.46DFT/CAMB3LYP w�w 4cDC TZ + 2s1p 154.75CCSD(T) ff 4cDC TZ + 2s1p 142.47CCSD(T) ff 4cDC⁄⁄ TZ + 2s1p 142.43CCSD(T) ffa 4cDC⁄⁄ TZ + 2s1p 142.26CCSD(T) ff 4cDC⁄⁄ QZ + 2s1p 142.19Exp. [27] 120.6(38)

a Energy threshold for virtual spinor: 134 Hartree.

Table 8Contact density gradient q½1�Pt

0 (in �4) of the PtSi molecule computed at various levelof theory. The derivative is calculated at the experimental equilibrium interatomicdistance Rexp

e ¼ 2:0615 Å [39].

Method qe Evaluation Hamiltonian Basis set q½1�Pt0

PtSiHF w�w 4cDC TZ �1168.75HF w�w 4cDC TZ + 2s1p �1183.83DFT/SAOP w�w NR TZ + 2s1p �135.64DFT/SAOP w�w X2C-sf TZ + 2s1p �2390.42DFT/SAOP hqPti X2C-sf TZ + 2s1p �732.09DFT/SAOP w�w 4cDC-sf TZ + 2s1p �729.65DFT/SAOP w�w 4cDC TZ + 2s1p �738.54DFT/LDA w�w 4cDC TZ + 2s1p �643.41DFT/PBE w�w 4cDC TZ + 2s1p �656.39DFT/BLYP w�w 4cDC TZ + 2s1p �625.41DFT/PBE0 w�w 4cDC TZ + 2s1p �789.17DFT/B3LYP w�w 4cDC TZ + 2s1p �728.07DFT/B3LYPa w�w DKH(2,0) ANO-RCC �2182.95DFT/B3LYPa w�w DKH(8,8) ANO-RCC �749.34DFT/CAMB3LYP w�w 4cDC TZ + 2s1p �837.70CCSD(T) ff 4cDC TZ + 2s1p �599.39CCSD(T) ff 4cDC⁄⁄ TZ + 2s1p �599.83

a M. Reiher and R. Mastalerz, private communication.

Table 7Contact density gradient q½1�Pb

0 (in �4) of the PbTe molecule computed at various levelof theory. The derivative is calculated at the experimental equilibrium interatomicdistance Rexp

e ¼ 2:5949 Å [93].

Method qe Evaluation Hamiltonian Basis set q½1�Pb0

PbTeHF w�w 4cDC TZ 236.84HF w�w 4cDC TZ + 2s1p 240.18DFT/SAOP w�w NR TZ + 2s1p 19.64DFT/SAOP w�w X2C TZ + 2s1p 585.06DFT/SAOP hqPbi X2C TZ + 2s1p 162.19DFT/SAOP w�w 4cDC-sf TZ + 2s1p 174.59DFT/SAOP w�w 4cDC TZ + 2s1p 161.41DFT/LDA w�w 4cDC TZ + 2s1p 148.16DFT/PBE w�w 4cDC TZ + 2s1p 155.74DFT/BLYP w�w 4cDC TZ + 2s1p 148.33DFT/PBE0 w�w 4cDC TZ + 2s1p 181.72DFT/B3LYP w�w 4cDC TZ + 2s1p 169.96DFT/CAMB3LYP w�w 4cDC TZ + 2s1p 197.55CCSD(T) ff 4cDC⁄⁄ TZ + 2s1p 163.19Exp. [27] 148(11)

S. Knecht, T. Saue / Chemical Physics 401 (2012) 103–112 109

2c h is obtained by block diagonalization U of the parent 4-compo-nent Hamiltonian 4c h

2ch ¼ ½Uy4chU�þþ: ð33Þ

Any 2-component one-electron property operator should be ob-tained by the same procedure, that is 2cX = [U�4cXU]++, rather thansimply taking the large–large (LL) block of the parent 4-componentoperator 2cX � [4cX]LL, an approximation that may lead to signifi-cant errors, in particular for properties probing electron densitynear nuclei (cf. Ref. [45]), as is the case here. The correct expressionof the electron charge density in some point P at the 2-componentrelativistic SCF level is

q2c ¼ �eXNocc

i

hw2ci j½U

ydðr� PÞU�þþjw2ci i – � e

XNocc

i

w2cyi ðPÞw

2ci ðPÞ

ð34Þ

Use of the untransformed operator, that is, the w�w expressionknown from the 4cDC and NR levels, leads to an overestimation ofq½1�Tl

0 in TlI by 283% at the SAOP/TZ + 2s1p level. Errors of similar sizeare observed for q½1�Pb

0 and q½1�Pt0 in PbTe and PtSi, respectively, show-

ing that for this property picture change errors are significant largerthan relativistic effects. X2C-calculations of the derivative densityq½1�X0 using the correctly transformed operator, (34), leads to errorsless than 1%, which is quite acceptable. In those calculations themajor source of deviation from the 4DC is probably the incompletetransformation of the 2-electron operator. For q½1�Pt

0 in PtSi resultsobtained with the Douglas–Kroll–Hess Hamiltonian are available[91]. At the DKH(2,0) level, that is, using a 2nd order DKH Hamilto-nian for the generation of orbitals and a 0th order (untransformed)property operator, we get picture change errors on the same orderas above, whereas at the DKH(8,8) level the deviation with respectto the reference B3LYP/4DC value is within 3%.

The effect of the basis set can be seen from the HF/4DC resultsobtained for q½1�Tl

0 in TlI. Starting from the TZ value of 209.30 �4,we see that adding tight 2s1p functions increases the value by1.4%, whereas going to the QZ + 2s1p level has only a minor effect.In the following we therefore restrict attention to results obtainedat the TZ + 2s1p level.

110 S. Knecht, T. Saue / Chemical Physics 401 (2012) 103–112

A first indication of the importance of electron correlation ef-fects is obtained by comparing SAOP and HF results obtained atthe 4cDC level. For all three molecules we observe that introductionof electron correlation through the asymptotically corrected SAOPfunctional reduces the HF value by 30–40%, clearly indicating thatthe inclusion of electron correlation is mandatory for this property.However, taking the CCSD(T)/4cDC⁄⁄ value as reference, we see thatthe performance of the various DFT functionals is variable andhardly satisfying. For TlI we see that LDA and GGA functionalsunderestimate the value of q½1�Tl

0 by some 20%, whereas the inclu-sion of exact exchange through hybrid functionals improves theresults somewhat. Interestingly, the long-range corrected func-tional CAMB3LYP shows quite different performance from that ofB3LYP. For q½1�Pt

0 in PtSi the hybrid functionals show the worst per-formance, and for q½1�Pb

0 in PbTe the simplest functional LDA agreesbest with the reference CCSD(T)/4cDC⁄⁄ value.

Finally, turning to the CCSD(T) results themselves, we see that forq½1�Tl

0 in TlI the molecular-mean field scheme 4cDC⁄⁄ introduces anegligible error compared to the full 4cDC calculation. In the formerapproach [81] all two-electron integral classes are employed in theSCF optimization step, but at the CC level only the (LLjLL) integrals,involving the large components only, are retained and employedin conjunction with the Fock matrix (orbital energies) obtained inthe SCF step. Comparing TZ + 2s1p and QZ + 2s1p values we further-more see that the value of the density derivative q½1�Tl

0 is wellconverged at the TZ + 2s1p level. For TlI we also investigated theconvergence with respect to the energy cutoff value for the virtualspace and found that the selected value introduces very small errors.

4.5. Comparison with experimentally derived data

We list in Table 9 for the three reference molecules our calculated4cDC⁄⁄-CCSD(T)/TZ + 2s1p contact density gradients in comparisonwith existing experimental and theoretical data. We furthermoreprovide estimates for the field shift parameter VX by inserting inEq. (28) the calculated contact density gradient along with theexperimental spectroscopic constants reported in Table 1. In passingwe note that if we combine our calculated data with the changes

Table 9Comparison of both the calculated and, where available, measured electron densitygradient q[1]X (in Å�4) at the Tl, Pb, and Pt nuclei and the field shift parameter VX (in104 Å�2). Besides our best theoretical estimates computed at the 4-componentmolecular mean-field CCSD(T)/TZ + 2s1p level (4cDC⁄⁄) we also list non-relativistic(NR) DFT/SAOP/TZ + 2s1p data.

Method Hamiltonian q[1]X VX

TlIExperiment[A]a 120.6(38) 3.20(10)Experiment[B]b 12.06(38)DFT/SAOP NR 21.09 0.57CCSD(T) 4cDC⁄⁄ 142.26 3.81DFT/SAOPc (ZORA) 21.1 0.61d

PbTeExperimenta 148(11) 2.12(16)Experimentb 14.8(11)DFT/SAOP NR 19.64 0.28CCSD(T) 4cDC⁄⁄ 163.19 2.36DFT/SAOPc (ZORA) 21.1 0.33d

PtSiexperiment[C]e �0.72(12)DFT/SAOP NR �135.64 �1.14CCSD(T) 4cDC⁄⁄ �599.83 �5.05DFT/SAOPe (ZORA) �136.5 �1.10

a Original result published in Ref. [27] in 1982.b Revised data published in Ref. [36] in 1985.c Ref. [38]; a QZ4P basis set was used in the DFT calculations.d Ref. [40].e Ref. [39]; a QZ4P basis set was used in the DFT calculations.

in mean-square nuclear radii dhr2ni from optical isotope shifts

(only available with unknown screening factor b of order unity)reported by Heilig and Steudel [8], we obtain changes in equilib-rium bond lengths for isotope pairs dRTlð203;205Þ

e ¼ �6:4 10�17 m;

dRPbð206;208Þe ¼ �3:9 10�17 m and dRPtð194;196Þ

e ¼ þ3:7 10�17 m, forTlI, PbTe and PtSi, respectively, which are exceedingly smallnumbers indeed.

Our point of departure for the following discussion will bethe experimental Tl(Pb) density gradients, q[1]Tl = 120.6 �4

(q[1]Pb = 148 �4), and field shift parameters, VTl = 3.20 � 104 �2

(VPb = 2.12 � 104 Å�2), which were reported by Schlembach andTiemann in the early 1980s [27] and are compiled in the rows‘‘experiment[A]’’ in Table 9. We find for both heavy metals a verygood agreement with our reference CCSD(T) values where we no-tice slightly larger deviations for the Tl nucleus. Scaling down thedensity gradient by 10%, as suggested by our study in Section 4.2on the validity of approximating the effective density by the con-tact density, would bring our CCSD(T) values in even closer agree-ment with ‘‘experiment[A]’’. Based on our re-derivation of theoryand our computed results we are thus confident to have provenwrong a suggested re-scaling of the ‘‘experiment[A]’’ densitygradients by a factor 10 which Knöckel and co-workers remarkedin a follow-up publication in 1985 [36] (denoted as ‘‘experi-ment[B]’’ in Table 9). More curiously, the authors obtained inthe latter work a VPb field shift factor for 208Pb32S which disagreesby approximately one order of magnitude with the first estimateby Schlembach and Tiemann!

Consider next the molecular field shift data derived fromDFT(SAOP)/QZ4P calculations by Cooke et al. [38]. Their valuesfor the contact density gradients in TlI and PbTe agree reasonablywell with the revised experimental data from 1985 [36], and thusseem to corroborate those. However, a comparison of their pre-dicted contact density gradient q[1]Tl = 21.1 �4 at the Tl nucleuswith our 4-component CCSD(T) value of q[1]Tl = 142.26 �4 revealsa rather striking discrepancy by more than a factor 6. We havedemonstrated in Section 4.4 that DFT density gradients cover arange relative to the CCSD(T) reference of up to 25% indicating thatthis cannot account for the wide variance of the DFT(SAOP)/QZ4P

value. A probable explanation may, on the other hand, be deducedfrom our non-relativistic (NR) contact density gradient and associ-ated field shift parameter data, which is for completeness added toTable 9 for all three diatomics. The evident agreement (apart fromminor basis set effects) of the two DFT(SAOP) data sets for eithermolecule strongly implies that Cooke and co-workers by accidentcarried out non-relativistic calculations though they were aimingfor scalar-relativistic studies based on the zeroth-order regularapproximation (ZORA); this has been confirmed for PbS [92]. Thisconclusion is further corroborated by recalling from Table 6 ourscalar-relativistic DFT(SAOP)/TZ + 2s1p contact density gradientat the Tl (Pb) nucleus q[1]Tl = 134.48 �4(q[1]Tl = 174.59 �4) which,inserted in Eq. (28) yields a field shift parameter VTl(VPb) on the or-der of 3.6 � 104 �2 (2.52 � 104 �2) rather than 0.61 � 104 �2

(0.33 � 104 Å�2). The close agreement of the nonrelativisticDFT(SAOP)/QZ4P data with the (presumably) erroneous lateexperimental results is merely fortuitous and this fact may explainwhy the computational mistake in Refs. [38–40] was left undiscov-ered to date.

With that said, we finally turn to the PtSi field shift parameterscompiled at the lower end of Table 9. Here, we see again a closematch of the contact density gradient q[1]Pt and field shift factorVPt derived from our non-relativistic DFT(SAOP) calculations andthe purportedly scalar-relativistic DFT(SAOP) calculations, whichfurther supports our present conclusions. The open questionremaining is then to explain the considerably large disagreementbetween our 4-component 4cDC⁄⁄ CCSDT (T)/TZ + 2s1p field shiftfactor VPt = �5.05 � 104 Å�2 and the experimental value of

S. Knecht, T. Saue / Chemical Physics 401 (2012) 103–112 111

�0.72 � 104 �2 reported by Cooke and co-workers [39]. Based onthe findings so far, we are confident of our theoretical estimate andtherefore propose a careful re-examination of the experimentalplatinum silicide data fit, in particular in the light of the non-con-forming Eqs. (25) and (26), the latter probably being employed inthe present data fit.

In summary, our extensive 4-component DFT and high-levelCCSD(T) reference data clearly shows by comparison with existingexperimental results for the contact density gradient and field shiftfactor in rotational spectra of the heavy-element diatomics TlI,PbTe, and PtSi that the original work of Schlembach and Tiemann[27] provided correct estimates for both properties. We further-more provide evidence that the revised experimental data ofKnöckel and co-workers [36] as well as more recent experimentaldata by Cooke et al. [39,40] are most likely incorrect and should bere-examined. We also conclude that previous DFT calculations [38–40], which appeared to support the more recent experimental val-ues, are seriously in error since they are accidentally based on anon-relativistic Hamiltonian.

5. Conclusions and perspectives

The objective of this study has been to provide an independentevaluation of both theoretical and experimental studies of the iso-topic field shift in rotational spectra using high-level relativistic 4-component electronic structure methods. The first experimentalevidence for an isotopic field shift effect in rotational spectra ofthe heavy-element Pb-chalcogenides and Tl-halides were reportedby Knöckel and Tiemann [26] and Schlembach and Tiemann [27]who identified it as ‘‘hidden’’ terms in the adiabatic mass-depen-dent correction to the Born–Oppenheimer (‘‘clamped nuclei’’)approximation (see for example [24]). The new atom-specific cor-rection terms were shown to be proportional to the gradient of theelectron (contact) density at a given nucleus with respect to theinternuclear distance [27], a quantity that can be extracted fromelectronic structure calculations. For the present benchmark studywe therefore chose three molecules TlI, PbTe, and PtSi, for whichexperimental and other theoretical data is available.

In the original formulation of the molecular isotopic field shift bySchlembach and Tiemann [27] the effective electron density at a gi-ven nucleus was identified as the contact density. We have investi-gated the validity of this approximation. Using a Gaussian modelof the nuclear charge distribution we find that the contact densityapproach yields a systematic overestimation of about 10% in allthree molecules compared to the electron density gradient derivedfrom the effective density, a deviation that can clearly not be ignoredin high-precision work. However, the systematic nature of the ob-served error suggests that it can be handled by a correction factor,although the implementation of the more accurate approach in acomputer code is straightforward and recommended.

Since heavy-element compounds are subject to relativisticeffects we have provided a thorough assessment of the reliabilityof different relativistic Hamiltonians for the calculation of thecontact density and its gradient. Our extensive calibration studiesshow that the inclusion of spin–orbit coupling in an exact-2-com-ponent or 4-component framework is mandatory if reasonablyhigh accuracy is aimed for. Whereas scalar-relativistic Hamiltoni-ans account for a major part of the relativistic effects with errorson the order of 10–12%, a non-relativistic ansatz yields meaninglesscontact density gradients for all three molecules reduced by up to afactor 6 compared to their 4-component reference values. Calcula-tions based on 2-component relativistic Hamiltonians show goodperformance, provided that the property operator is correctlytransformed. Otherwise we find that the picture change errorsare larger than relativistic effects. We furthermore point out the

importance of considering electron correlation effects in the prop-erty evaluation by comparing results both from a choice of densityfunctionals and high-level CCSD(T) calculations with Hartree–Fockdata. As summarized recently in a related study on the isomer shiftin mercury fluorides [35] density functional theory (DFT) with var-ious flavors of functionals exhibits a rather inconclusive overallperformance relative to our CCSD(T) reference for the contactdensity gradient, covering not only a range of ±25% but showingalso a strong system-dependency based on the quality of a resultfor a given functional type. One of the appealing feature of DFT isof course its low computational cost and it may thus, nevertheless,be applicable to further qualitative studies of field shift effects inother heavy-element diatomics. In this context, we have demon-strated how detailed information about changes in electronicstructure upon changes in internuclear distances can be extractedfrom HF/DFT expectation values by projection analysis.

We finally evaluated the known experimental and theoreticalelectron density gradients and field shift factors for the three nu-clei Tl, Pb, and Pt on the basis of our benchmark 4-componentCCSD(T) data. The comparison holds a surprising twist in the taleand may be best summarized as follows:

1. Our present 4-component CCSD(T) electron density gradientsand field shift factors are in very good agreement with theexperimental predictions given by Schlembach and Tiemann[27] in 1982 with slightly larger deviations for Tl in TlI thanPb in PbTe.

2. As a consequence our results do not agree at all with the sug-gested scaling of the original data from 1982 which was pub-lished in 1985 by Knöckel et al. [36] along with a revisedformula to derive the field shift factor from a fit of rotationalspectra data. We have carefully re-derived the appropriate for-mulas and cannot see that such a scaling is justified, nor wereany arguments provided for it in Ref. [36].

3. On the other hand, previous DFT(SAOP) predictions [38,39] ofelectron density gradients and field shift parameters of therespective heavy-metal centers in the present three moleculesagreed quite well with the revised experimental data byKnöckel and co-workers [36]. We conclusively show, though,that these calculations are plagued by a serious computationalerror. Having aimed at relativistic studies based on the ZORAHamiltonian, these calculations turned out to be of non-relativ-istic nature as a comparison with our present non-relativisticDFT(SAOP) data unambiguously reveals. It so happens, and thisis the twist in the tale, that the relativistic effect is of about thesame order as the scaling factor proposed by Knöckel et al. [36]to be applied to the original and correct data by Schlembach andTiemann [27], which is probably why these inconsistencieswere not discovered previously.

4. Our predicted field shift parameter for Pt in PtSi differs by a fac-tor 7 from the experimental value reported by Cooke and co-workers [39] which we currently attribute to be caused by amistake in the experimental data fit. We therefore recommenda careful re-examination of experiment.

We hope that our re-derivation of the key equations accompa-nied by the computational findings will put the theory of molecu-lar isotopic field shift in rotational spectra back on solid footingsand stimulate further experimental and theoretical work in thisfield. In future work we aim at investigating nuclear size effectsof vibrational spectra, in which the second geometrical derivativeof the effective/contact density comes into play. We will alsocontinue projection analysis of these quantities to see what infor-mation about molecular electronic structure they contain. Ulti-mately it will be interesting to see if a combination of state ofthe art correlated electronic structure calculations combined with

112 S. Knecht, T. Saue / Chemical Physics 401 (2012) 103–112

experiment can be used to extract information about changes innuclear volume upon isotope substitution.

Acknowledgements

We dedicate this paper to Debashis Mukherjee for his profoundinsight into the electron correlation problem and for his manyimportant contributions on how to treat it. S.K. gratefully acknowl-edges postdoctoral research grants from l’ Université de Strasbourg(UdS) and the Natural Science Foundation (FNU) of the DanishAgency for Science, Technology and Innovation. We would like tothank Markus Reiher and Remigius Mastalerz at ETH Zürich forproviding us their unpublished results on the molecular field shifteffect in PtSi. We would furthermore like to thank Radovan Bast(Toulouse) for graphical assistance. This work has been supportedthrough ample computing time at the supercomputer centers ofUdS and the Danish Center for Scientific Computing at SDU Odense.This work is part of the project NCPCHEM funded by the AgenceNationale de la Recherche (ANR, France).

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