Nuclear motion effects on the density matrix of crystals: An ab initio MonteCarlo harmonic approachCesare Pisani, Alessandro Erba, Matteo Ferrabone, and Roberto Dovesi Citation: J. Chem. Phys. 137, 044114 (2012); doi: 10.1063/1.4737419 View online: http://dx.doi.org/10.1063/1.4737419 View Table of Contents: http://jcp.aip.org/resource/1/JCPSA6/v137/i4 Published by the American Institute of Physics. Additional information on J. Chem. Phys.Journal Homepage: http://jcp.aip.org/ Journal Information: http://jcp.aip.org/about/about_the_journal Top downloads: http://jcp.aip.org/features/most_downloaded Information for Authors: http://jcp.aip.org/authors

THE JOURNAL OF CHEMICAL PHYSICS 137, 044114 (2012)

Nuclear motion effects on the density matrix of crystals: An ab initioMonte Carlo harmonic approach

Cesare Pisani, Alessandro Erba, Matteo Ferrabone, and Roberto DovesiDipartimento di Chimica IFM and Centre of Excellence NIS (Nanostructured Interfaces and Surfaces),Università di Torino, via P. Giuria 5, I-10125 Torino, Italy

(Received 21 May 2012; accepted 2 July 2012; published online 27 July 2012)

In the frame of the Born-Oppenheimer approximation, nuclear motions in crystals can be simulatedrather accurately using a harmonic model. In turn, the electronic first-order density matrix (DM) canbe expressed as the statistically weighted average over all its determinations each resulting from aninstantaneous nuclear configuration. This model has been implemented in a computational schemewhich adopts an ab initio one-electron (Hartree-Fock or Kohn-Sham) Hamiltonian in the CRYSTAL

program. After selecting a supercell of reasonable size and solving the corresponding vibrationalproblem in the harmonic approximation, a Metropolis algorithm is adopted for generating a sampleof nuclear configurations which reflects their probability distribution at a given temperature. Foreach configuration in the sample the “instantaneous” DM is calculated, and its contribution to theobservables of interest is extracted. Translational and point symmetry of the crystal as reflected inits average DM are fully exploited. The influence of zero-point and thermal motion of nuclei onsuch important first-order observables as x-ray structure factors and Compton profiles can thus beestimated. © 2012 American Institute of Physics. [http://dx.doi.org/10.1063/1.4737419]

I. INTRODUCTION

Detailed information about the electronic structure ofcrystalline compounds is provided by observables related tothe one-electron density matrix (DM),1 such as the electroncharge density (ECD) [ρ(r)], and the electron momentumdensity (EMD) [π (p)]. The ECD is obtainable from diffrac-tion experiments and is straightforwardly related to the topo-logical features of the system in direct space, thus to positionof nuclei and characteristics of bonds. The EMD can be recon-structed from directional Compton scattering experiments:2

the analysis of the distribution in momentum space of the slowvalence electrons is known to provide valuable complemen-tary insight into the chemical features of the system.

Quite recently, the entrance into a new age of molecularquantum chemistry has been declared: “In the fourth age weare able to incorporate into our quantum chemical treatmentthe motion of nuclei [...] and compute accurate, temperature-dependent, effective properties, thus closing the gap betweenmeasurements and electronic structure computations.”3 Theapproach presented here goes in that direction and allows forthe description, even if approximate, of nuclear motion effectson the electronic density matrix and related properties of crys-talline materials.

Rarely diffraction or Compton scattering experiments arecarried out at sufficiently low temperature such that the effectof thermal nuclear motion can be completely neglected. Dueto the fact that core and inner-valence electrons of atoms fol-low the movement of the respective nuclei, when ECD and re-lated x-ray structure factors (XSF) are considered, it is manda-tory to account for the effect of finite temperature, for in-stance by means of atomic harmonic Debye-Waller thermalfactors.4 An enormous amount of literature has been devotedto the approximate evaluation of such effects in order to al-

low for a correct interpretation of the x-ray scattering data.5

In particular, attempts to go beyond the harmonic approxima-tion and to properly describe thermal diffuse scattering havebeen presented extensively.6–9 The rigid-atom approximation,which involves an arbitrary partition into atomic subregionsof the total ECD of the system, is, however, usually retained.The approach that we present in this paper allows to go be-yond such a questionable, even if effective in most cases,approximation.

The situation is completely different as concerns theother fundamental DM-related property, EMD: in this casethe usual approximation of completely neglecting nuclear mo-tions can result in a reasonable estimate of the Compton pro-files (CP) and the effect of finite temperature is seldom ex-plicitly considered, although few ad hoc models have beenreported.10, 11 After recalling that the content of information ofa rich set of directional CPs is the same as that of the EMD,12

the following question can be asked: why is the distributionof electron velocities so scarcely affected by the movementof the nuclei? The answer is physically intuitive. The velocityof core electrons is hardly affected by the movement of theirrespective nuclei; that of valence electrons is more sensitiveto the instantaneous change of the distances between neigh-boring nuclei, but the effect is in a sense a second-order one.It is therefore to be expected that rather sophisticated compu-tations are required in order to unambiguously recognize theresponse of CPs to nuclear motion. We shall show in Sec. IIIthat the present method actually can.

The information contained in the DM of a crystal, whichis complementary to that concerning total energy and relatedquantities, is extremely helpful for assessing merits and lim-itations of computational schemes for crystals. We have re-cently shown that very accurate directional CP, as can be

0021-9606/2012/137(4)/044114/11/$30.00 © 2012 American Institute of Physics137, 044114-1

044114-2 Pisani et al. J. Chem. Phys. 137, 044114 (2012)

measured from the inelastic scattering of high intensity syn-chrotron radiation by single-crystal samples,13, 14 can revealsubtle aspects of the electronic structure of periodic systems.In the cases of Urea,15 Silicon16, 17 and Quartz18 the theo-retical CPs obtained using single-determinantal approxima-tions to the ground-state wave function were found in fact topresent definite discrepancies with respect to the experiment,which were partly removed when use was made of an ab initiotechnique based on a multi-determinantal description of thewave function.19–21 This interesting outcome has encouragedus to take a closer look to the possible reasons for the residualdifferences between theoretical and experimental CPs. In thispaper we address one of the aspects which are usually disre-garded when simulating EMDs that is, the effect of zero-pointand thermal motion of nuclei on DM-derived quantities. Theeffect of temperature on both the ECD and the EMD is treatedhere on the same footing.

Before describing analytically in Sec. II the modeladopted, let us anticipate its essentials. The Born-Oppenheimer (BO) approximation will be systematicallyadopted, according to which the eigenfunctions �M (R) andthe associated eigenvalues EM which describe the nuclearmotions result from the Schrödinger equation for the nucleiin the potential field V (R); in turn, V (R) is the ground-stateenergy of the many-electron system corresponding to nucleifixed in configuration R. The Boltzmann probability distri-bution for the nuclei at a temperature T can be expressed asfollows:

PT (R) = 1

Z(T )

∑M

exp(−EM/kB T ) |�M (R)|2 ;

(1)Z(T ) =

∑M

exp(−EM/kB T ) ,

where kB is Boltzmann’s constant and Z(T) the nuclear parti-tion function. By invoking again the BO approximation, thestatistically averaged DM will be given by

γ T (r; r′) =∫

dR PT (R) γ (r; r′|R) . (2)

This expression looks formidable, in particular because theintegrand is extended in principle to the nuclear coordinatesof the whole crystal, and translational symmetry cannot be ex-ploited, except for a trivial invariance of PT (R) with respectto a translation of all nuclear coordinates by the same latticevector. We shall show below that the use of the harmonic ap-proximation combined with a supercell (SC) model will makethe evaluation of γ T computationally feasible.

Practically, the approach to be described here has beenimplemented in the ab initio public program CRYSTAL,22, 23

which solves the “periodic” problem for a number ofone-electron self-consistent-field Hamiltonians. Either theHartree-Fock (HF) approximation, or one of a variety ofdensity functional theory (DFT) schemes in the Kohn-Sham formulation, or still hybrid-exchange techniques canbe adopted.24, 25 The extension of the present approach topost-HF techniques is discussed with reference to the publicCRYSCOR program.19, 20, 26, 27

In this paper we present the formalism of a new ab initioMonte Carlo approach to the rigorous study of nuclear motioneffects on a variety of DM-related properties of crystallinematerials, both in position and momentum space. The influ-ence of the main parameters of the method is investigated,with reference to the simple test case represented by the cu-bic crystal of silicon, in order to demonstrate its accuracy andconsistency; preliminary results on both x-ray structure fac-tors and CPs are presented.

The plan of this paper is as follows. Section II exposesthe procedure adopted for evaluating the thermally averagedobservables and it is divided into six subsections. Section IIA presents and justifies the SC-harmonic model of nuclearmotions which gives us an explicit expression of PT (R). Sec-tion II B describes the Metropolis algorithm here adopted forgenerating a set {RI } of nuclear configurations distributed ac-cording to PT: the statistical significance of the sampling setand the choice of a suitable “Metropolis step” is discussedtheoretically and numerically taking crystalline silicon as atest case. Section II C demonstrates how it is possible to ex-ploit the full symmetry of the crystal which is preserved inthe average observables, thus increasing by a large factor thesignificance of the sampling set. In Sec. II D the computa-tional set-up adopted in CRYSTAL for solving the harmonicproblem and collecting preliminary information is discussedin general terms. Section II E shows how the DM associatedto each given nuclear configuration of the sampling set canbe obtained from SC calculations performed with CRYSTAL.In Sec. II F, we consider the possibility of approximately cor-recting the statistical DM obtained from a Hartree-Fock (HF)calculation in order to include correlation effects by usingthe CRYSCOR program,19, 20, 26, 27 which implements Møller-Plesset perturbation theory truncated at order two (MP2) in alocal,28 fully periodic formulation. In Sec. III the above pro-cedure is applied to crystalline silicon and some preliminaryresults are reported as concerns the computation of x-ray dy-namical structure factors and thermal CPs. The intent is hereto investigate the effect of the main parameters involved onthe computed properties. Some general conclusions are drawnand an outline of prospective work is sketched in Sec. IV. Ex-plicit formulae for the determination of the expansion coeffi-cients of the DM observables and details about their compu-tational implementation are provided in the Appendix.

II. THEORETICAL DETERMINATION OF THESTATISTICAL DENSITY MATRIX

A. Supercell-harmonic description of nuclear motions

The quantum-mechanical description of the nuclear mo-tions in the BO approximation is dictated by the shape ofthe potential energy surface V (R). Let us fix for definitenesssome notations. The configuration R which describes the in-stantaneous position of all nuclei in the crystal can be spec-ified by assigning to each of them the displacement with re-spect to their equilibrium position,

R ≡ {. . . , [(R0)a + g + dga] , . . .} . (3)

044114-3 Pisani et al. J. Chem. Phys. 137, 044114 (2012)

The index a (a = 1, . . . , N) labels the general nucleus in theunit cell and (R0)a is its equilibrium position in the refer-ence zero cell; the lattice vector g = ∑3

m=1 lgm am identifies

the general crystal cell where am are the direct lattice vectors:in a cyclic crystal model the integers l

gm run from 0 to Lm − 1.

When dealing with cubic crystals (as here for simplicity) allLm’s are set to a common value L. V (R) is a function of thedisplacements dg

a or, equivalently, of their cartesian coordi-nates x

gj , the index j running from 1 to 3N. By expanding in a

Taylor series V (R) with respect to these coordinates aboutthe equilibrium R0 configuration, after setting V (R0) = 0,and exploiting translational invariance, the usual expression isobtained

V (R) = L3

2

∑g

∑i,j

Vgij x0

i xgj + O3

({x

gj

}). (4)

The harmonic approximation which will be adopted here con-sists in neglecting all O3 terms. This truncation to second-order terms implies that the potential energy only dependson the independent (non-coherent) motion of two atoms at atime; it however does not entail, as we shall see, that the sameproperty holds true for the resulting probability distributionPT (R).

Before using the consequences of the harmonic approx-imation, it is expedient to justify briefly its use in the presentcontext. Its enormous success is due not only to its simplicitybut also to its predicting power. The evaluation of the second-order derivatives with respect to the displacement coordinatesaround the equilibrium configuration (V g

ij ) can nowadaysbe performed accurately and cheaply by standard periodiccodes based on one-electron Hamiltonians. The solution ofthe corresponding Schrödinger equation is straightforward,as we shall recall in a moment. The accurate descriptionof vibrational frequencies for most crystalline compoundswhen suitable Hamiltonians are adopted is a strong argumentin its favor. Its main deficiencies in this respect are theneed for anharmonic corrections when very light atomsare involved and for long-range terms to account for thelongitudinal-optical/transverse-optical (LO/TO) splitting inionic compounds. More generally relevant is the fact thatin the harmonic approximation the equilibrium positions ofatoms do not change with temperature (see below).

It is shown in standard textbooks29 that translational in-variance permits us to factorize the harmonic Schrödingerequation for the nuclear motions into L3 separate ones, eachassociated to a wavevector k = ∑3

n=1 (κn/L) bn where bn arethe reciprocal lattice vectors and the integers κn run from 0to L − 1. After defining for each k the 3N × 3N dynamicalmatrix Wk, with elements

W kij = 1√

MiMj

∑g

exp(ık · g) Vgij

(Mi being the mass of the nucleus associated to the ith coordi-nate), the solution is obtained through the diagonalization ofWk

(Uk)† Wk Uk = �k. (5)

The elements of the diagonal �k matrix provide the vi-

brational frequencies νki =

√λk

i (here and in the following

atomic units are adopted), while the columns of the Uk matrixgive the corresponding normal coordinates

qki = 1√

L3

3N∑j=1

√Mj Uk

ji

∑g

exp(ık · g) xgi .

The general eigenfunction and the corresponding eigenvalueare identified by the vector of non-negative integers M

= {. . . , mk1, . . . , m

k3N, . . .}, which assigns the level of excita-

tion of all vibrational modes (phonons):

�M (R) =∏i,k

Hmki

[(λk

i

) 14 qk

i

], (6)

EM =∑i,k

εmki

[εmk

i=

(mk

i + 1

2

)νk

i

2 π

]. (7)

Hm(x) is proportional to the normalized associated Her-mite function of order m; the correspondence R ←→ {xg

i }←→ {qk

i } ←→ {ξki } ≡ {(λk

i )14 qk

i } is implicit here and in the fol-lowing. ξ are the frequency-scaled coordinates; ξ = 1 corre-sponds to the classical elongation.

Substitution in Eq. (1) and exploitation of the simple de-pendence of EM on the excitation levels, results in the follow-ing compact expression of the probability distribution for thenuclei in the harmonic [(h)] approximation:

P(h)T (R) =

∏i,k

′pT (ξk

i ),

(8)

pT (ξki ) =

[1 − exp

(− νk

i

2π kB T

)]×

∑mk

i

exp

(− mk

i νki

2π kB T

) ∣∣Hmki

(ξki

)∣∣2.

Note that the zero of energy has been set here at the funda-mental harmonic level E0. The product

∏′ excludes the threezero-frequency modes which correspond to pure translations.From P(h)

T the average value of the product of a given multipletof displacement coordinates can be obtained after integrationover all remaining coordinates. Trivially, 〈xg

i 〉(h)T = 0 for all

coordinates: only the inclusion of anharmonic corrections canaccount for their temperature dependence, in particular for thechange of volume with temperature. Of special interest is the(3N) × (3NL3) rectangular matrix X which gives the averagevalue of pair-products (due to translational invariance, the firstcoordinate can be assigned to the zero cell)

Xgij = 〈x0

i xgj 〉(h)

T . (9)

Here 〈· · ·〉(h)T denotes the expectation with respect to the har-

monic probability density of the displacements at temperatureT.30 The 3 × 3 blocks along the diagonal of the square sub-matrix {X0

ij } define the so-called atomic displacement param-eters (ADP) which are used in the analysis of x-ray factors.The X matrix can also be calculated analytically in the har-monic approximation.31

044114-4 Pisani et al. J. Chem. Phys. 137, 044114 (2012)

B. Sampling the nuclear configurations

Having at hand the explicit though approximate expres-sion (8) for the harmonic distribution of nuclear motions,we want to generate from there a set of coordinates {RI }(I = 1, . . . , I ) such as to provide a non-biased estimate ofthe statistically averaged DM as defined in Eq. (2) whose ac-curacy will depend on the size I of the sample

γ(h)T (r; r′) ≈ 1

I

I∑I=1

γ (r; r′|RI ). (10)

A Metropolis algorithm32 will be adopted for this purposewhich can be described as follows, r representing each time anew random number from a uniform distribution in the inter-val [0, 1].

1. Choose a temperature T and select the size I of the sam-ple and a step τ of exploration of the normal coordinates.Set the current I value to 1. Choose a starting configura-tion RI ≡ {ξk

i }I (for instance, set initially to 1 all nor-mal coordinates) and calculate the corresponding valuePI = P(h)

T (RI ) from Eq. (8).2. Generate a new trial configuration Rt by adding to each

normal coordinate of the Ith step the quantity (2r − 1)τ ;calculate P t = P(h)

T (Rt ).3. If Pt/PI < r , then “reject the step forward” and back-

substitute the trial configuration with the current one(Rt = RI , Pt = PI).

4. Set I to I+1 and define: RI = Rt , PI = Pt. If I < I , thengo to step 2 for a new trial; otherwise exit the procedure.

This kind of selection is simple and rapid. Before dis-cussing the reliability and usefulness of the collected sample{RI }, let us define an equivalent weighted coordinate sample(WCS):

{{RJ , wJ }} ≡ {RI }. (11)

Here the index J = 1, . . . ,N labels each newly acceptedconfiguration, while the weight wJ is the number of timesthat configuration is repeated at steps 3-4 of the Metropolisprocedure before accepting a new one. N is thus the actualnumber of accepted configurations. Equation (10) can then berewritten as follows:

γ(h)T (r; r′) ≈ 1

I

N∑J=1

wJ γ (r; r′|RJ ). (12)

The average weight w = I/N obviously increases with in-creasing τ (it is less likely with a large exploration step tomove from a reasonable configuration to another of compa-rable probability). It has originally been suggested, for themaximum significance of a sample of size I , to select τ insuch a way that w ≈ 2, that is that half of the trials are ac-cepted on average.33 However, a lot of work has been doneto calibrate the optimal acceptance rate of such algorithm inits several formulations.34 Here, the cost of the computationis proportional to N (number of single-point self-consistent-field calculations) rather than to I , so we are interested in hav-ing relatively large values of w. However, as shown below, the

step τ cannot be increased indefinitely without deterioratingthe statistical significance of the sample.

The remaining part of this section is devoted to the dis-cussion of how to choose reasonable values for N and τ ; asa test case, we consider the crystal of silicon. In order to de-couple the nuclear motions of atoms formally assigned to dif-ferent unit cells in the crystal, the use of a supercell (SC) isnecessary, as far as systems with small unit cells are consid-ered. In the case of crystalline silicon, for instance, a repetitiveSC corresponding to L = 2 (with reference to the conventionalcubic cell of this face-centered-cubic crystal), which contains64 atoms, is essentially adequate.31 This corresponds to 3NL3

− 3 = 189 normal modes to be sampled.In order to appreciate the significance of a given set

of accepted weighted nuclear configurations (defined by Nand τ ), we can evaluate from it selected multiplet-products〈 〉 ≡ 〈x0

i xgj 〉 of displacement coordinates and associate to

them their corresponding standard deviations σ about theiraverage 〈 〉av . For infinite sampling size all σ ’s should bezero. To define standard deviations σ , once fixed the valuesof N and τ , the Metropolis algorithm is repeated 1000 timesso that, at the end, we are left with 1000 determinations of〈 〉 from which an average 〈 〉av and a corresponding stan-dard deviation σ are easily obtained. Tables I and II clearlyshow how such average values are stable.

Table I shows the dependence on temperature of two im-portant pair-products 〈 〉av and of their standard deviationsfor I ≈ 3000 and for four values of τ : 0.1, 0.2, 0.3, and 0.4.The values there reported are obtained with the PBE gener-alized gradient functional36 of the DFT and a 6-21G* basis

TABLE I. Average (〈 〉av) and standard deviation (σ ) of two pair-products of displacement coordinates for silicon as a function of temperature,in 10−4 Å2: a diagonal element 〈x0

1 x01 〉, namely the anisotropic displacement

parameter (ADP), and the most important out-of-diagonal element whichrefers to the simultaneous displacement of two nearest neighbor atoms in theSi-Si-Si plane 〈x0

1 xg1 〉. The standard deviations concern sets of weighted con-

figurations of approximatively the same size (I ≈ 3000) but obtained withfour different τ values, as indicated; the respective average weights w are re-ported. These results refer to the PBE functional with the 6-21G* basis setfrom Ref. 35 on a conventional SC with L = 2.

〈 〉av σ

T (K) 〈x01 x0

1 〉 〈x01 x

g1 〉 〈x0

1 x01 〉 〈x0

1 xg1 〉 τ w

0 21.026 3.192

⎧⎪⎪⎨⎪⎪⎩0.0770.0960.1470.300

0.0610.0810.1340.285

0.40.30.20.1

41.110.9

3.81.7

100 23.841 4.892

⎧⎪⎪⎨⎪⎪⎩0.0970.1170.1760.343

0.0820.1050.1560.332

0.40.30.20.1

33.89.73.61.7

200 33.263 8.604

⎧⎪⎪⎨⎪⎪⎩0.1420.2150.3170.574

0.1310.2090.3040.563

0.40.30.20.1

21.37.33.11.6

300 45.995 13.352

⎧⎪⎪⎨⎪⎪⎩0.2390.3420.4861.028

0.2230.3350.4721.017

0.40.30.20.1

13.85.62.71.5

044114-5 Pisani et al. J. Chem. Phys. 137, 044114 (2012)

TABLE II. Standard deviations σ ’s related to the average 〈 〉av

= 〈x01 x0

1 〉av pair-product of displacement coordinates (i.e. to the ADP) atT = 298.15 K as a function of the adopted Metropolis step τ and number ofaccepted configurations N . Computational parameters as in Table I.

N

100 500 1000 10000 100000 τ w

17.631 0.729 0.514 0.015 0.008 0.1 1.58.202 0.373 0.274 0.074 0.005 0.2 2.7

45.973 ± 4.813 0.246 0.174 0.065 0.002 0.3 5.64.067 0.190 0.139 0.059 0.002 0.4 13.83.324 0.150 0.100 0.048 0.001 0.5 42.2

set from Ref. 35, on a conventional SC with L = 2. The firstpair-product in this table represents the atomic anisotropicdisplacement parameter (ADP) of crystalline silicon whichis here obtained as a by-product of the scheme. The valueat 300 K (45.99 × 10−4 Å2), for instance, can be comparedto the corresponding analytical determination of it: 45.98× 10−4 Å2.31 The following can be observed.

The average weight w of each set of weighted config-urations decreases with temperature, as expected, and it isabout an order of magnitude larger with the larger τ . A stepτ ≈ 0.1 - 0.2 corresponds to the standard Metropolis choice,since the weight in that case is about 2 at all temperatures.The much better performance of the sets of weighted config-urations obtained using τ = 0.4 is evident from the fact thatthe corresponding σ ’s are smaller by about a factor 5 withrespect to the case τ = 0.1. It can incidentally be noted thatthose standard deviations increase with temperature, and thatfor given T, τ their value is practically the same for the twopair-products.

As it is known from standard statistical analysis, the stan-

dard deviations σ ’s are proportional to 1/√

I . The size of thesample required for a given accuracy can be estimated fromthat dependence and from the data of w reported in Table I.Let us recall that the size of the sample I = N × w definesthe number N of self-consistent-field calculations to be per-formed on a supercell without any symmetry (apart from thetranslational one at the SC scale). In order to reduce the com-putational resources needed and to make this technique fea-sible at all, the lowest number of configurations compatiblewith a chosen accuracy has to be selected.

To drive such a choice, we report in Table II the standarddeviations σ ’s related to the 〈 〉 = 〈x0

1 x01 〉 pair-product of

displacement coordinates (i.e., to the ADP of silicon) at T =298.15 K as a function of the adopted step τ and number ofaccepted configurations N . Computational parameters are asin Table I. It is seen than the standard deviations are reducedboth by increasing the number of accepted configurations Nand increasing the step τ . Since the computational cost is pro-portional to N , it is expedient to exploit the dependence fromτ . From the table it can be inferred that, even with 500 ≤ N≤ 1000, good accuracies (≈ 0.3%) could be reached by usinghigh values for the step; for τ = 0.5 the average weight w ofeach configuration in Eq. (12) turns out to be 42.2 that is veryconvenient in this case.

TABLE III. Average square of the frequency-scaled normal coordinates〈ξ2〉(N , τ, T ) in a sample of N = 1000 configurations, as a function of theadopted step τ at two temperatures: 0.001 K and 298.15 K.

T (K)

τ 0.001 298.15

0.1 0.4988 0.87340.2 0.5012 0.87540.3 0.4989 0.87230.4 0.5019 0.87100.5 0.4993 0.87240.6 0.5031 0.87310.7 0.5087 0.87660.8 0.5163 0.87980.9 0.5274 0.88451.0 0.5315 0.88871.1 0.5362 0.89301.2 0.5389 0.8972

Unluckily, as mentioned before, step τ cannot be in-creased indefinitely without deteriorating the significance ofthe statistical sample. For instance, given a particular choiceof temperature T, number of accepted configurations N andstep τ , let us define the quantity

〈ξ 2〉(N , τ, T ) = 1

N3NL3

N∑J=1

wJ

∑ik

∣∣ξki,J

∣∣2. (13)

This is a measure of the average square of the frequency-scaled normal coordinates in the sample. When T = 0 K,all the 3NL3 phonons in the crystal are found in their funda-mental state and 〈ξ 2〉(N , τ, 0K) should be theoretically equalto 1/2. In Table III, we report this quantity as a function ofthe adopted step τ , for a sample of N = 1000 configurationsat two temperatures: 0.001 K and 298.15 K. The data forT = 0.001 K clearly show a diverging deviation from the ex-pected value of 1/2 for τ ≥ 0.7. A similar divergence is ob-served also in the data at 298.15 K.

At the end, we have seen how, to obtain a reliable statisti-cally averaged DM according to Eq. (12), the electronic prob-lem for the supercell comprising NL3 atoms should be solvedabout a thousand times (N ≈ 103). The additional difficultymust be considered that no symmetry is retained by the gen-eral RI configuration except the translational one at the SCscale. Space group symmetry can however be exploited whileperforming the averaging procedure, as described in Sec. II C.

C. Full exploitation of symmetry

The general symmetry operator (S) of our cyclic crystaland its effect on any function of the space coordinates can beexpressed as follows:

S = {VS | gS + t(VS)},S [f (. . . , r, . . . , r′, . . .)] = f (. . . , rS, . . . , r′

S, . . .), (14)

rS = (VS)−1r − gS − t(VS).

Here V is a 3×3 matrix which effects a point symmetryoperation on the r coordinates; there are h different suchmatrices which form on the whole the point group of the

044114-6 Pisani et al. J. Chem. Phys. 137, 044114 (2012)

crystal. g = ∑3m=1(lgm mod L) am is the general lattice vec-

tor of the cyclic crystal, while t(V ) is a fractional transla-tion which may be associated to a subgroup of V rotations(in symmorphic space groups, all t(V )’s are 0). The order ofthe cyclic space group is then S = h L3. Each sampled RI is infact representative of S equivalent configurations. When eval-

uating the thermal average Y(h)T of a quantity of interest Y, this

property can be used to reduce the sample size from N (asit would be required if no account of symmetry were taken)to N /S for a given significance. Let us see now how this ispossible with reference to the two most important DM-relatedquantities, namely the ECD, ρ(r), and the EMD, π (p) (see theIntroduction).

The scheme below summarizes the relation betweenthese quantities and the DM in the coordinate and momen-tum representation, related to each other by a six-dimensionalFourier transformation (6D-FT):

γ (rS, r′S) = γ (r, r′)

D−→ ρ(r) = ρ(rS)

� 6-FT(15)

γ (pV , p′V ) = γ (p, p′)

D−→ π (p) = π (pV )

γ (−p,−p′) = γ (p, p′)D−→ π (p) = π (−p) .

HereD−→ means taking the diagonal of a two-variable func-

tion, while pV = V −1 p . The relation in the last line is due totime reversal symmetry, and is valid even if the point groupdoes not contain the inversion. Note the different symmetryfeatures of ECD and EMD for the perfect crystal, which mustbe preserved when performing the thermal average.

A symmetry-adapted form of the thermally averagedECD is simply obtained by its expansion in an orthonormalset of symmetrized plane waves (SPW) {�s(r)}, associatedto the different stars of K vectors of the reciprocal lattice.Each star is characterized by a representative vector Ks, andby a subgroup of V rotations (V i

s , i = 1, hs) which generateall members of the star (Ki

s , i = 1, hs) when applied to Ks.We then can write (W is the volume of the primitive cell):

ρT (r) =∑

s

As�s(r) =∑

s

As

{hs∑

i=1

ωsi exp(ıKi

s · r)

};

(16)

ωis = 1

hs

√W

exp[ − ıKs · t

(V i

s

)].

Note that the As coefficients are precisely the structure factorsrelated to the thermal ECD. While a large number of termsmust be considered if we want to accurately reproduce all finedetails of the ECD (in particular the distribution of core elec-trons), we are actually interested in the effect of nuclear mo-tions on the low-index x-ray structure factors; so perhaps thefirst few hundred terms in the expansion are sufficient.

The statistically averaged EMD can be expressed in asymmetry-adapted form as follows:

πT (p) =∑j,�s

Cj,�s �j,�s(p),

(17)�j,�s(p) = fj (p) Z�s(θ, φ).

Here fj(p) is a suitable set of radial functions, while Z�s(θ , φ)are symmetrized combinations of spherical harmonics (SSH)of angular quantum number �. They can be obtained by firstapplying the totally symmetric projection operator of thepoint group I = ∑

V V to the general Y�m(θ , φ), then normal-izing the resulting functions and eliminating redundancies.Note that in the formation of the projector I, we must sup-plement the operators of the original point group with thoseresulting from their multiplication by the inversion, if the lat-ter is not among them (see Eq. (15)). For this same reason,only SSH of even � can contribute.

In the Appendix we shall provide explicit expressionsfor the evaluation of the expansion coefficients in Eqs.(16) and (17) in the case where the instantaneous DM is theresult of a calculation performed using a one-electron Hamil-tonian and a basis set of Gaussian type orbitals (GTO).

D. The solution of the harmonic problem

All calculations to be analyzed in the following concernthe self-consistent solution of one-electron equations for a pe-riodic non-conducting system described by a nuclear config-uration R:

hX ψXi (r) = εX

i ψXi (r) , (18)

where X labels a one-electron Hamiltonian. From there, boththe ground-state energy V X[R] and the DM are obtained:

γ (r; r′|R) =∑

i

′2 ψX

i (r)[ψX

i (r′)]�

. (19)

In these equations i is a composite label for the crystallineorbitals ψX

i , which includes a wave-vector and a band index;∑i′ runs over all solutions belonging to the occupied mani-

fold, that is to the Nel/2 lowest bands, Nel being the number ofelectrons in the unit cell.

All such one-electron calculations are performed here bymeans of the CRYSTAL program, whose characteristics andcapabilities are described elsewhere.22, 23 It is just expedient torecall that the same basis set is adopted as in standard molec-ular codes that is, the crystalline orbitals are expressed as alinear combination of GTOs. By this conventional name wemean a “contraction” of Mμ normalized Gaussian primitivesof angular momentum components �, m centered in RA:

χμ(rA) =Mμ∑j=1

djμ Gm

�

(αj

μ; rA

),

(20)Gm

� (α; r) = Nm� (α) X�,m(r) exp[−α r2] .

Here rA = r − RA, Xm� are real solid harmonics and Nm

�

normalization coefficients; djμ are known as “contraction co-

efficient”, αjμ as “exponent” of the general primitive which

contributes to the χμ GTO. We shall adopt here the stan-dard choice of taking as RA the nuclear coordinates, even ifCRYSTAL allows the use of GTOs centered in “ghost atoms”at a general position.

The quality (and cost) of a CRYSTAL calculation dependsbasically on the choice of the effective Hamiltonian hX andon that of the GTO basis set. In order to analyze this issue,

044114-7 Pisani et al. J. Chem. Phys. 137, 044114 (2012)

let us recall the information which is required by the presentscheme.

The vibrational problem must first be solved as describedin Sec. II A for obtaining the probability distribution for thenuclei, P(h)

T (R). This kind of calculations concern the perfectsystem [R = R0 ≡ {. . . , [(R0)a + g] , . . .} , see Eq. (3)],so the power of the CRYSTAL code as regards the use of sym-metry is fully exploited. The requirement is that the equilib-rium configuration R0 and the Hessian matrix of the secondderivatives of the energy about it, {V g

ij }, be accurately de-scribed. As anticipated in Sec. II A the solution of the vibra-tional harmonic problem also permits us to select an appropri-ate SC size for the sampling of the displacement coordinates.

These preliminary calculations which concern the perfectsystem also allow us to collect some quantities to be used forperforming the integrals in Eqs. (A1) and (A8).

If a post-HF correction has to be performed, CRYSTAL

must provide the reference HF solution for CRYSCOR: thisissue will be dealt with in Sec. II F.

E. One-electron supercell calculations

Totally different calculations are needed for obtaining theDM determinations γ (r, r′|RJ ) from which the statisticallyaveraged ECD and EMD are derived. They are expected to bevery cumbersome because they involve SCs with NL3 atomsand no point symmetry. Furthermore, their number (N ) maybe very large (N ≈ 103).

In any event, under the hypotheses adopted, the SC char-acterized by the super-unit-vectors Am = Lam coincides withthe repetitive cell. This means that the crystalline orbitalsfrom Eq. (18) corresponding to a given RJ configurationhave the full translational periodicity of the cyclic crystal:ψJ

i (r) = ψJi (r − Lg). In other words they are associated to

the zero wavevector �, and the “band index” i fully identifiesthem.

From the representation of the crystalline orbitals in theGTO basis set, and from Eq. (19), we then have

ψJi (r) =

∑μ

aJμ,i

∑g

χμ

(rJA − Lg

),

γ (r, r′|RJ ) =∑μ,ν

P Jμ,ν

×⎧⎨⎩∑

g,g′χμ

(rJA − Lg

)χν

(r′J

B − Lg′)⎫⎬⎭ , (21)

P Jμ,ν = 2

∑i

′aJ

μ,i aJν,i .

In the sums above, μ (ν) runs over the GTOs centered in allthe L3 atoms A (B) in the supercell. Note that the basis setis different for each RJ : in fact, even if all atoms equivalentto the reference one in the primitive cell (A ∼ a, see Eq. (3))share the same set of GTOs as defined in Eq. (20), their center(R0)a + g + dg

a depends on the specific configuration. Due tothe short range of the GTOs and to the relatively large sizeof the SC, the sums over g, g′ can be limited to the first starof lattice vectors about the origin, if we are interested in r, r′

coordinates within the SC or in its neighborhood. The primedsum in the last line runs over the Nel L3/2 occupied orbitals;their real character at � has been considered.

The ECD is immediately obtained as the diagonal ele-ment of the DM: ρ(r|RJ ) = γ (r, r|RJ ). Consider now theEMD. The Fourier transform of the GTO χμ(rJ

A) can be writ-ten as χF

μ (p) exp[ıp · (rJ

A)], where the factor χF

μ (p) is theFourier transform of the μ-th GTO centered in the origin, andis therefore independent of J, and the same for all A ∼ a. Us-ing the definition (15) and Eq. (21), we then have

π (p|RJ ) =∑μ,ν

cJμ,ν(p) χF

μ (p)[χF

ν (p)]∗

,

(22)cJμ,ν(p) =

∑A∼a, B∼b

P Jμ,ν exp

[ıp · (

rJA − rJ

B

)].

F. Post-Hartree-Fock correction to the DM

We recalled in the Introduction that going beyond amono-determinantal description of the DM seems criticallyimportant especially as concerns observables such as the CPswhich are related to the distribution of electron velocities.A preliminary analysis is here performed about the possi-bility of including in a unified model the effects of nuclearmotions accounted for with the harmonic scheme just pre-sented, and those of electron correlation treated with a multi-determinantal post-HF ab-initio technique. Let us admit thatthe harmonic problem can be solved with an appropriate one-electron Hamiltonian at a reasonable level of accuracy (seeSec. II D); we then concentrate on the correlation correctionto the DM. We shall consider for this purpose the prospectiveuse of our periodic local MP2 code CRYSCOR,19, 20, 27 whichhas been applied successfully to the simulation of the DM ofcrystalline systems.15–18, 37

The full exploitation of the crystalline symmetry, theclever use of the local-correlation Ansatz28 combined withpowerful density fitting techniques21, 38 for the evaluation oftwo-electron integrals, make the CRYSCOR calculations rel-atively cheap; in particular, quasi-linear scaling of the com-putational cost with the “size” of the problem is achieved, tobe compared to the (Nel)5 scaling of standard schemes basedon the use of canonical orbitals.21, 27 The main outcome of aCRYSCOR calculation is the MP2 energy E(2). Our main inter-est here, however, is in the correlated DM which is obtainedfrom the excitation amplitudes owing to a Lagrangian tech-nique that is, by evaluating the first-order orbital-unrelaxedresponse of the energy of the system to an arbitrary externalone-electron perturbation.15, 39, 40

In principle, such correction should be applied to each in-stantaneous DM γ HF(r, r′|RJ ) resulting from a CRYSTAL SCcalculation (see Eq. (21)). We would so obtain for each con-figuration RJ a corrected P matrix in the corresponding basisset: P J,HF

μ,ν −→ P J,MP2μ,ν . The expansion coefficients of the ther-

mally averaged observables including the correlation correc-tion could then be obtained by simply feeding such correctedP matrix in the expressions provided in Sec. II E.

While formally straightforward, this scheme is prac-tically unfeasible at present because it would require avery large number of SC CRYSCOR calculations. All the

044114-8 Pisani et al. J. Chem. Phys. 137, 044114 (2012)

simplifications due to exploitation of point symmetry arethere lost, and the cost would be prohibitively high.

A much simpler scheme can be tentatively explored. TheMP2-DM is first evaluated for the equilibrium configurationR0, which has the full symmetry of the perfect crystal

P 0,MP2μ,ν = P 0,HF

μ,ν + P 0,corrμ,ν .

From P0, corr the correlation corrections to the “static” HF ob-servables (that is, by neglecting nuclear motion) can be ob-tained. These corrections are then simply summed to the ther-mally averaged ones computed using the HF Hamiltonian.This approximation is equivalent to summing the same cor-rective P0, corr matrix to the general PJ, HF one, by formallyidentifying χμ(r0

A − Lg) with χμ(rJA − Lg):

P J,MP2μ,ν ≈ P J,HF

μ,ν + P 0,corrμ,ν . (23)

The idea behind this simplified scheme is that, while the ef-fects on the DM of the changes of internuclear distances (pri-marily, Pauli repulsion effects) due to nuclear motions are es-sentially taken into account at the one-electron level, thoseassociated to the instantaneous Coulomb repulsion betweenelectrons of different spin are less critically dependent on thenuclear coordinates. It can be observed that in the frame ofthis approximate scheme the basis set employed for the deter-mination of P0, corr and related observables could (and should)be much richer than that used in the SC HF calculations.

III. PRELIMINARY RESULTS

In this section we present some preliminary results whichare intended to investigate the effect on computed one-electron properties of the number N of nuclear configurations(i.e. of single-point self consistent field calculations) consid-ered. We consider the cubic crystal of silicon and we com-pute x-ray structure factors Fhkl (the consolidated set of 19reflections of Ref. 41) and a directional CP Jhkl(p) (along thecrystallographic direction [hkl] ≡ [111]). It is shown belowthat the effect of temperature (298 K) is clearly revealed bythis approach as a function of N . Computed x-ray structurefactors are compared to their experimental determinations.

This scheme has been implemented in the “periodic”ab initio CRYSTAL program that is here used.22, 23 All quanti-ties of interest in the program are expressed as linear combi-nations of Gaussian “primitive” functions centered in high-symmetry positions: these functions, atomic orbitals, AO,constitute the so-called basis set (BS). We start from a 6-21G*BS from Ref. 35 which was also used in a recent study of theECD of crystalline silicon (BS1).1 We further enrich this BSwith three single-primitive polarization functions, one of dand two of f type with exponents of 0.3, 0.8, 0.2 a.u., respec-tively. In CRYSTAL, the truncation of infinite lattice sums iscontrolled by five thresholds, which are here set to 8,8,8,8,16.The Hamiltonian used for both the lattice dynamics and thecalculation of the one-electron properties is the PBE func-tional of the DFT.36 A shrinking factor of 2 is adopted whichcorresponds to 8 k-points in the irreducible Brillouin zone.

In the case of crystalline silicon, a repetitive SC corre-sponding to L = 2 (a conventional cubic cell of this fcc crystalwhich contains 64 atoms) is essentially adequate. This corre-

FIG. 1. Percentage differences RXhkl = (FX

hkl − Fexp

hkl )/F exp

hkl × 100 of dy-namical structure factors of silicon at 298 K as computed with the PBE func-tional and the basis set described in the text. The effect of the number N ofnuclear configurations used in Eq. (12) is also reported. N = 1 correspondsto the static case.

sponds to 3NL3 − 3 = 189 modes to be sampled. A detaileddiscussion of the effect of the number of nuclear configura-tions N and step τ is addressed in Sec. II B. It is there demon-strated that values of τ = 0.5 and 500 ≤ N ≤ 1000 providean accurate description of the nuclear motion effects on com-puted one-electron properties both in position and momentumspaces.

We first report results obtained for the dynamical struc-ture factors of silicon at 298 K with the PBE functional, up toN = 1000. Given the computational parameters above, eachof these calculations takes 2.5 h, in average, running in paral-lel on eight processors on a IBM-SP6 cluster at the CINECA

supercomputing center. We have considered up to 103 nuclearconfigurations for a total CPU time of ∼2 × 104 h.

In Figure 1 we report the percentage differences RXhkl of

dynamical structure factors of silicon at 298 K as computedwith the Monte Carlo scheme illustrated above, with the PBEfunctional. The effect of the number N of nuclear configura-tions used in Eq. (12) is also reported (note that N = 1 corre-sponds to the static limit). From the analysis of this figure thefollowing observations can be inferred: (1) there is a clear andstrong dependence on N of the computed values; (2) globally,the results appear almost converged for N > 500; (3) as ex-pected, the effect of N is more evident for high hkl indices re-flections which are indeed the most dumped by temperature;(4) the intensity of the “forbidden” 222 reflection, whose dif-ference from zero gives a measure of the asphericity of theelectron charge distribution around the silicon atoms, is theonly one that grows with temperature.

For N = 1000, the agreement with the experiment can bedescribed by an overall agreement factor of R = 1.83%. Tack-ing into account the many approximations involved in such ascheme, an overall agreement lower than 2% is to be consid-ered a nice evidence of its effectiveness. Let us note that herethe computed values overestimate the experiments (i.e., thedynamical correction is undershot). This might suggests (afterconsidering the intrinsic limitations of the adopted Hamilto-nian) that some of the thresholds involved in the Monte Carlo

044114-9 Pisani et al. J. Chem. Phys. 137, 044114 (2012)

FIG. 2. Differences between the 298 K and 0 K directional CPs of siliconalong the [111] direction. The CP at 298 K is computed from the average overN nuclear configurations [J (p;N )] while that at 0 K from a single nuclearconfiguration [J(p; 1)].

technique are still too poor as, for instance, the size of theadopted supercell or the number N of nuclear configurationsadopted. As a consequence, a systematic way for further im-proving the description of thermal nuclear motion within sucha scheme exists.

As regards CPs, a direct comparison with the experimentwill be possible at a further stage since experimental data areunder analysis; in Figure 2 we report the differences betweenthe directional CPs of silicon along the [111] crystallographicdirection computed from the statistical average over N nu-clear configurations J (p;N ) at 298 K and that computed ona single (equilibrium, 0 K) nuclear configuration J(p; 1). Asdiscussed before for structure factors, also in this case the ef-fect of N is quite evident and gives rise to a peculiar oscillat-ing behavior of the CP.10, 11 Again, the computed results arefound to be almost converged for N ≈ 500. To be fair, even ifunambiguously revealed, the effect of thermal nuclear motionon the total CPs of crystalline silicon is here found to be quitesmall (within 0.1%) so that very accurate measurements arerequired for an experimental detection of it. The group of Y.Sakurai has been carrying out these experiments at two tem-peratures (10 and 300 K) at the synchrotron center of SPring-8 in Japan and their results will be processed, analyzed and,finally, compared to ours and discussed in the next future.

To conclude, these preliminary results show that the ap-proach illustrated in this paper is well-behaved as concernsits internal consistency and can definitely reveal the effect ofthermal nuclear motion on computed one-electron propertiesof crystals in a fully ab initio fashion.

IV. CONCLUSIONS

A new approach for the ab initio study of thermal nu-clear motion effects on a variety of one-electron properties ofcrystalline materials is presented which consists in the defi-nition of a statistically averaged first-order density matrix via

a Monte Carlo technique for the sampling of the harmoniclattice potential within Born-Oppenheimer approximation.

Both translational and point symmetries are fully ex-ploited at each step of the procedure and a detailed discussionof this topic is reported.

This technique, which is rather general, allows for the rig-orous study of the effect of temperature on the electron chargedensity, x-ray structure factors, Compton profiles, and elec-tron momentum density of a crystal. In particular, as concernsx-ray structure factors, it allows to go beyond the commonlyadopted rigid-atom approximation which consists in arbitrar-ily partitioning the total electron charge density into subre-gions that can be associated to different atoms and in keep-ing such partition also when nuclear motions are taken intoaccount (Debye-Waller factors are computed within such anassumption).

The preliminary results presented are intended to demon-strate that this approach can definitely reveal the effect of ther-mal nuclear motion on computed one-electron properties ofcrystals. This technique will be applied to the simulation oftemperature effects on momentum-space directional Comp-ton profiles of covalent crystalline silicon and ionic lithiumfluoride for which accurate Compton scattering experimentshave been carried out at 10 K and 300 K by our collaboratorsY. Sakurai and M. Itou at the synchrotron center of SPring-8in Japan and the collected data will be analyzed in the nextfuture.

ACKNOWLEDGMENTS

We acknowledge the CINECA Award No.HP10AX8A2M, 2011 for the availability of high per-formance computing resources and support.

APPENDIX: EXPANSION COEFFICIENTS: THEFORMULAE

In this appendix we provide explicit expressions for thedetermination of the expansion coefficients of ρT (r) andπT (p) in terms of symmetrized functions, as defined inSec. II C. We shall assume that the instantaneous DMγ (r, r′|RJ ) is available in a GTO representation as discussedin Secs. II E and II F.

Due to the orthonormality of symmetrized plane-waves(SPW), the expansion coefficients of ρT as resulting from aweighted coordinate sample (WCS) {{RJ , wJ }} are simplygiven by [use is made of Eqs. (16) and (21)]

As =∑

J

wJ

NL3

{∑μ,ν

P Jμ,ν

∑g,g′

hs∑i=1

ωsi

×∫

dr χμ

(rJA − Lg

)χν

(rJB − Lg′) exp

[ıKi

s · r]}

.

(A1)

In Appendix C of Ref. 42 it is shown how to evaluate an-alytically the integral following the strategy proposed byMcMurchie and Davidson43 that is, by using as an expansion

044114-10 Pisani et al. J. Chem. Phys. 137, 044114 (2012)

set Hermitian Gaussian Type Functions (HGTF)

�[p, P; t, u, v] =(

∂

∂x

)t (∂

∂y

)u (∂

∂z

)v

exp(−p |rP|2) ,

(A2)where rP = r − P ≡ (x − Px, y − Py, z − Pz). The productof two GTOs defined as in Eq. (20), with angular momentumquantum numbers � and �′, respectively, can be expressed asa linear combination of HGTFs:

χμ(rA) χμ′(rA′) =∑

jj ′,tuv

F [μ, j ; μ′, j ′; t, u, v]

×ZAj,A′j ′ �[pjj ′ , Pjj ′ ; t, u, v] ,

(A3)

where

pjj ′ = αjμ + α

j ′μ′ , Pjj ′ = (

αjμA + α

j ′μ′A′)/pjj ′ ,

ZAj,A′j ′ = exp( − α

jμα

j ′μ′ |A − A′|2/pjj ′

). (A4)

The indices j, j′ run over all primitives of the two GTOs, whilet, u, v are triplets of integers such that t + u + v = � + �′.The F coefficients can be obtained owing to recursion rela-tions as explained in Ref. 42; they are independent of RJ andcan thus be calculated once and for all. The integral in Eq.(A1) thus reduces to a linear combination of Fourier trans-forms of HGTFs:∫

dr �[p, P; t, u, v] exp(ıK · r)

=∫

dx

(d exp[−p (x − Px)2]

d x

)t

exp(ıKx x)

×∫

dy

(d exp[−p (y − Py)2]

d y

)u

exp(ıKy y)

×∫

dz

(d exp[−p (z − Pz)2]

d z

)v

exp(ıKz z)

= 8(πp)3/2 (ıp)t+u+v exp(ıK · P) exp [−K2/(4p)]. (A5)

While exploring the WCS, the quantity in curly brackets inEq. (A1) is thus easily obtained for each s, and progressivelysummed up to finally provide the thermally averaged structurefactors.

The extraction of the expansion coefficients of the EMD(Eq. (17)) follows a different pattern. It is first necessary tospecify the {�j, �s(p)} set. Our choice has been to construct itby starting from an auxiliary set G of “Gaussian primitives”defined as in Eq. (20) but in momentum space, and all cen-tered in the origin: G ≡ {Gm

� (α; p)}; they are precisely theproduct of a radial function by a spherical harmonics. Afterapplying to each of them the projector I, eliminating redun-dancies and normalizing the result as specified in Sec. II C,we are left with n symmetrized functions:

�j,�s(p) =∑m

aj,�s;m Gm� (αj ; p) . (A6)

Because of the inversion symmetry, only even �’s are needed.We have verified that using a relatively restricted G set withsuitably chosen “exponents” (αj) allows a faithful repro-

duction of the momentum distribution of typical crystallinesystems. This choice provides us with an analytic form ofthe thermally averaged EMD, very convenient for subse-quent manipulations. On the other hand, it entails the non-orthonormality and, possibly, the linear dependency of the ex-pansion set. A standard technique is adopted to get rid of thisproblem. After obtaining by a unitary transformation (U† S U= �) eigenvalues and eigenvectors of the n × n overlap S ma-trix (whose evaluation is straightforward), we can eliminatelinear dependencies by excluding from � rows and columnsrelated to eigenvalues below a pre-fixed threshold (� −→ �),and from U the corresponding columns (U −→ U ). The n × m

matrix A = U �−1/2

allows us to define an orthonormal setof m ≤ n functions:

FV (p) =∑j,�s

Aj,�s;V �j,�s(p) . (A7)

Reference can be made formally to this {FV (p)} set for ob-taining a convenient expression for the expansion coefficientsof the thermally averaged EMD:

(C(h)j ′,�′s ′ )T =

∑j,�s;V

Aj ′,�′s ′;V Aj,�s;V I j,�s ,

where

I j,�s = 1

N∑

J

wJ

∫dp �j,�s(p) π (p|RJ ) . (A8)

The integral in the last line has to be evaluated for each givennuclear configuration by using the corresponding expressionof the instantaneous EMD, Eq. (22). Even if its value can beobtained analytically, a numerical technique has been adoptedfor this purpose. It takes advantage of the experience gained inperforming integrals over atomic domains of different kindsof functionals of the electronic density as are required in DFTcalculations44–46 (note however that in this case no partitioninto atomic contributions is required). The present scheme isthe same as implemented in CRYSTAL22 and reproduces withfew differences the one proposed in Ref. 46. Essentially, aEuler-MacLaurin-Lebedev grid is adopted comprising Np “ra-dial” points pn with weight wrad

n and, for each pn, Nω “angu-lar” points (θω, φω), with weight w

angω . Actually, it is often

found convenient to adopt different values of Nω for differentradial values (so-called “pruning” technique).

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