Evaluation of the electron momentum density of crystalline systems from ab initio linear combination...

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Evaluation of the Electron Momentum Density of CrystallineSystems from Ab Initio Linear Combination of AtomicOrbitals CalculationsAlessandro Erba∗[a] and Cesare Pisani†[a]

Alternative techniques are presented for the evaluation ofthe electron momentum density (EMD) of crystalline systemsfrom ab initio linear combination of atomic-orbitals calcula-tions performed in the frame of one-electron self-consistent-fieldHamiltonians. Their respective merits and drawbacks are ana-lyzed with reference to two periodic systems with very different

electronic features: the fully covalent crystalline silicon and theionic lithium fluoride. Beyond one-electron Hamiltonians, a post-Hartree–Fock correction to the EMD of crystalline materials is alsoillustrated in the case of lithium fluoride. © 2012 Wiley Periodicals, Inc.

DOI: 10.1002/jcc.22907

Introduction

Because of recent advances in both experimental and theoret-

ical techniques, a lot of attention has been devoted in the last

years to the electron momentum density (EMD) of crystalline

systems. It is known in fact that the EMD, π(p), can provide

valuable information on the electronic structure of the system,

complementary to that embodied in the electron charge distri-

bution (ECD), ρ(r). Although the latter function can be obtained

rather routinely from X-ray diffraction measurements, the EMD

can be reconstructed from a rich set of very accurate directional

Compton profiles.[1, 2] The comparison of this kind of experimen-

tal data with the results of computer simulations can be very

fruitful. On one hand, it helps us to interpret the features of

the EMD, in particular its anisotropy, in terms of the bonding

structure of the crystal[3–7]; on the other hand, it may reveal

definite inadequacies of the theoretical model adopted, which

are not apparent when considering observable quantities that

only depend on the total energy or on the ECD.[8–10]

It is important, therefore, that computer codes that simulate

the electronic properties of crystalline systems can provide on

request detailed information on the EMD and related properties.

This article is concerned with this issue. More specifically, we will

present and critically compare different techniques for extracting

values of π(p) at a selected set of points P ≡ {p} in momentum

space, corresponding to a single-detor description of the crystal

ground-state wavefunction:

�0(x1, . . . , xN) = A[ψ1(x1) . . . ψN(xN)].

The crystalline orbitals (COs) entering this antisymmetrizedproduct are obtained from the self-consistent solution of theSchrödinger equation: hX ψX

i= εX

iψX

i, where hX is an effec-

tive ab initio one-electron Hamiltonian, the apex X specifyingits type: Hartree–Fock (HF), Kohn–Sham Density-Functional The-ory (KS-DFT), or hybrid exchange. We shall also assume thatCOs are expressed as a linear combination of atomic orbitals(AOs); more details about their analytic expression are provided

in the following. �0 solutions of this kind are provided forinstance by the CRYSTAL code, and use will be made below ofthis well-tested program[11, 12] to demonstrate the performanceof the various techniques considered. It is not our intention todiscuss here the limitations of such single-detor approximationsof the wavefunction as concerns their ability to describe EMD-related properties; reference can be made for this purpose toprevious work.[8, 9, 13–15] However, we will show how a correlationcorrection, as obtained for instance using the post-HF CRYSCOR

code,[16–19] can be superimposed onto the πHF(p) determinationof the EMD.

In Section Computational Techniques, the mathematical back-ground is briefly recalled for the sake of definiteness. Even if theapparatus adopted is quite standard, some subtle problems arisein applications, which need unambiguous definitions for theiranalysis. Two formally equivalent expressions for the calculationof the EMD are next presented. The corresponding compu-tational techniques perform very differently, both concerningaccuracy and cost, according to the specific problem they areapplied to. A special but important task, that of obtaining 2Dmaps of the EMD of 3D crystals, is treated in some detail.

Section Examples of Application provides examples of theperformance of the different techniques, with reference to twosimple crystalline systems: a covalent (silicon) and an ionic(lithium fluoride) one.

Computational Techniques

A list of the main symbols to be used in what follows is reported,along with a description of what they represent, in Appendix A.

[a] A. Erba, C. Pisani

Dipartimento di Chimica IFM,Centre of Excellence

NIS (Nanostructured Interfaces and Surfaces),Università di Torino,

via P.Giuria 5, I-10125 Torino, Italy

E-mail: [email protected]

†Deceased.

© 2012 Wiley Periodicals, Inc.

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Themodel: conventions and settings

The direct and reciprocal lattice vectors of our D-dimensionalcrystal are defined by the basis vectors {ai} and {bj}, respectively,satisfying ai · bj = 2 π δij (i, j = 1, . . . ,D).

A cyclic-crystal model is adopted by introducing D integers{si}, which define super-basis vectors {Ai = si ai}: the cyclic-ity condition corresponds to assuming that �(. . . , rnωn, . . .) =�(. . . , (rn + G)ωn, . . .), that is, the wavefunction is unalteredfrom any direct-space translation by a super-lattice vectorG = ∑

i Gi Ai (integer Gi ’s). The L = ∏D

i si inequivalent lat-tice vectors g = ∑

i gi ai with {0 ≤ gi < si} (integer gi ’s) willbe said to constitute the G set.

Correspondingly, the shrunk-basis vectors of the reciprocalspace {bj = bj/sj} define a Monkhorst net: κ = ∑

j κj bj (integerκj ’s). The L vectors of this net with {0 ≤ κj < sj} will be said toconstitute the K set. Equivalently, each κ ∈ K can be made tobelong to the first Brillouin zone of the crystal, by displacing itclosest to the origin through translation by a reciprocal latticevector.

The relationships are known to hold true, to be used below(K is a general vector of the reciprocal lattice):

∑g∈G

exp[ ı(κ − κ ′ − K) · g ] = L δκκ ′ [κ , κ ′ ∈ K]∑κ∈K

exp[ı(g − g′ − G) · κ] = L δgg′ [g,g′ ∈ G] (1)

to describe the one-electron states of our crystal, which com-prises n electrons per cell, we shall use a basis set (BS) consistingin Lp translationally equivalent, localized real functions, theAOs, identified by a Greek letter (µ) running from 1 to p

and by the crystal cell g ∈ G in which they are centered:|µg〉 ←→ χ

gµ(r − rµ). AOs are in turn a linear combination of

Gaussian type orbitals (GTOs). Their explicit expression and theirFourier transform (FT) are provided in Appendix B.

From the set {|µg〉}, an equivalent set of Lp Bloch functions(BF) can be formed, associated with points κ ∈ G: {|µκ〉 ←→φ

κµ(r)}:

|µκ〉 = 1√L

∑g∈G

exp(ı κ · g)∑G

|µ(g + G)〉, (2)

The p BFs at a given κ are basis functions for the κth irre-ducible representation (irrep) of the translational group of thecrystal and satisfy the cyclicity condition [φκ

µ(r) = φκµ(r + G)].

A one-electron Hamiltonian hX [�] is now considered, whichdepends on the one-electron density matrix (DM) �, as it isspecified below. We will assume for simplicity hX [�] to bespinless, that is, to act identically on α- and β-spin elec-trons; the generalization to spin-dependent Hamiltonians isstraightforward.

In the BF BS, hX is block diagonal:

〈µκ|(hX [�])|νκ ′〉SC = δκκ ′Hκµν (3)

Here and below, the subscript SC means that the integral isrestricted to r values within the general supercell of the cycliccrystal.

The p CO at the general κ : {|iκ〉 ←→ ψκ

i(r)} are obtained by

solving the p × p matrix equations:

Hκ Cκ = Sκ Cκ Eκ ; (Cκ )† Sκ Cκ = I, (4)

where Sκµν = 〈µκ|νκ〉SC, Eκ

ij= ε

κ

iδij , and I is the identity matrix.

We have:

|iκ〉 =∑

µ

Cκ

µi | µκ〉 , or:

ψκ

i(r) =

∑µ

Cκ

µi

∑g∈G

exp(ı κ · g)∑G

χg+Gµ (r − rµ)

Because of Eq. (4) and to the orthogonality of functionsbelonging to different irreps, we easily have:

〈iκ|jκ ′〉SC = δij δκκ ′ . (5)

The Fermi energy EF can now be introduced, which permitsus to define an “occupied set” O comprising exactly nL/2 COs|iκ〉 such that ε

κ

i< EF. That is, O contains the nL spin orbitals

lowest in energy [�κ

i(x) ≡ ψ

κ

i(r) α(ω), �

κ

i (x) ≡ ψκ

i(r) β(ω)],

which can host all electrons of the repetitive unit of the cycliccrystal (note that nL is assumed to be an even integer, whereasthis may not be true for n).

The ground-state wavefunction of the cyclic crystal is:

�0(x1, . . . , xnL) =∥∥∥∥∥∥

∏iκ∈O

[�

κ

i(x2I−1) �

κ

i (x2I)] ∥∥∥∥∥∥ , (6)

where ‖ ∏[ξi(xi)]‖ is the normalized Slater determinant (detor)obtained from an orthonormal set of spin orbitals, whereas I

(running from 1 to nL/2) labels the occupied COs.The corresponding spinless DM in a direct-space representa-

tion is:

γ (r, r′) = 2∑iκ∈O

ψκ

i(r)

[ψ

κ

i(r′)′]∗

=∑µν

∑gg′∈G

Pg−g′µν

∑GG′

χg+Gµ (r − rµ)χg′+G′

ν (r′ − rν)

(7)

Pgµν = 2∑iκ∈O

Cκ

µi

(C

κ

νi

)∗exp[ı κ · g] (8)

This DM is in principle the one which should enter self-consistently in the definition of hX [�] when HF and KS theoriesare considered. More precisely, its diagonal part, ρ(r) = γ (r, r)determines the Hartree potential and the exchange-correlationpotential in KS-DFT Hamiltonians, whereas its out-of-diagonalpart defines the exchange operator in the HF Hamiltonian:K f (r) = ∫

dr′γ (r, r′) f (r′)/|r − r′|.

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EMD expressions

The EMD is the diagonal part of the 6D FT of the direct-spaceDM[20, 21]:

π(p) = L−1

∫SC

dr dr′ exp [ı p · (r′ − r)] γ (r, r′) (9)

We are now introducing two equivalent expressions of the EMD,to be referred to as the “P-formula” and the “C-formula” in thefollowing.

The P-formula. When performing the integral of eq. (9), advan-tage can be taken of the simple expression of the FT of GTO-AOs(see Appendix B). We then have, using eqs. (7) and (B2):

π(p) =∑µν

∑g∈G

Pgµν exp [ıp · (rν − rµ − g)] χµ(p) [χν(p)]∗ (10)

where χµ(p) is the FT of the AO χµ(r) centered on the originof the spatial coordinates. The factor L−1 and the limitation ofthe integral to the supercell in eq. (9) have allowed us to setto zero g′, G, G

′. With this convention, π(p) is normalized to

n, the number of electrons in the unit cell.Equation (10) requires preliminarily the definition of a

Monkhorst net and of the corresponding K set. At all symmetry-independent κ points among those L ones, the Hκ matrix mustbe reconstructed and diagonalized [eqs. (3) and (4)]. The Pg

matrix must next be evaluated, according to eq. (8), at all vectorsg ∈ G. The Fermi energy determined in the self-consistent-field(SCF) procedure (ESCF

F ) can be used in this step, or its valuerecalculated.

As a matter of fact, the Pg matrix used in the SCF part of theCRYSTAL code is obtained differently. Independently of the inte-gers si that define the Monkhorst net, the [Pgµν]SCF elements arecalculated over a symmetric set of lattice vectors g ∈ Gµν ⊂ Gspecific of the two AOs involved, and dictated by an overlap cri-terion between them; substantial savings in computer time andstorage are so achieved. We shall call this procedure pseudo-overlap (po) truncation.[22] From now on, we shall generallyassume that this truncation is not affecting the properties ofthe DM, although in one instance we shall demonstrate howthe accuracy of the calculated EMD may depend on the potolerances adopted in the self-consistent part of the procedure(see Section The Pseudo-Overlap Truncation Tolerance).

The use of the P-formula now permits the EMD to be eval-uated at any p. Note, however, that it provides the exact valueonly at points of the extended Monkhorst net (p = κ +K) thatis at points that coincide with a point of the K set except for atranslation by a reciprocal lattice vector. At intermediate points,it provides the corresponding Fourier interpolation based onthe G set; this can be a quite good estimate if the Monkhorstnet is sufficiently dense and the dependence of π(p) on prather smooth. Particularly, critical is the case of metals wherethe EMD is discontinuous at the Fermi surface.

The use of the P-formula is also needed when one wantsto estimate the effect of correlation on the DM, hence also onthe EMD, using a post-HF program like CRYSCOR.[16] This code

provides in fact an estimate of the correction (Pg,corrµν ) evaluated

by means of a perturbative Møller–Plesset technique truncatedat second order (MP2).[8, 17, 18, 23] This must be added to P

g,HFµν to

obtain the correlated DM in an AO representation. The set ofthe corrected P

gµν matrices can be directly used in eq. (10).

TheC-formula. Consider now the case where the point p = κ+Kbelongs to the extended Monkhorst net. Substitution in eq. (10),use of the definition (8) of P

gµν and of the property (1) gives:

π(p) = 2∑µν

∑iκ ′∈O

Cκ ′µi

(C

κ ′νi

)∗exp [ıp · (rν − rµ)]

×∑

g∈Gexp [ı(κ ′ − κ − K) · g]

χµ(p) [χν(p)]∗

=∑µν

Pκµν exp [ıp · (rν − rµ)] χµ(p) [χν(p)]∗ (11)

Pκµν = 2

∑i

′C

κ

µi

(C

κ

νi

)∗ [p = κ + K] (12)

The sum (∑

i

′) in the last line runs over the occupied COs at

the κ point.Equations (11) and (12) allow for the computation of the

EMD π(p) only at p points of the extended Monkhorst netcorresponding to κ points where the Hκ matrix has been actuallydiagonalized and its use is then preferable when the exact valueof the EMD is required at few selected p points in space, asin the case of an EMD map in a given plane (see Section 2DEMD Maps).

To clarify the differences between the two expressions (P-and C-formulae), let us consider the extreme case where thevalue of the EMD is needed in just one particular point of theextended Monkhorst net: π(p�) with p� = κ�+K�. The C-formulawould then require the diagonalization of just one matrix (Hκ�

),whereas the P-formula would require the diagonalization of thewhole set of L matrices Hκ with κ ∈ K for the definition of thePg matrix in eq. (8).

Furthermore, the use of the C-formula is particularly rec-ommended when EMD features of metallic systems have to beevaluated. Note, however, that the use of ESCF

F is here mandatory,as this technique does not provide an independent estimateof the Fermi energy.

2D EMDmaps

A computationally convenient scheme for drawing a 2D map ofthe EMD of a 3D crystal in a selected plane is now described insome detail, which uses the C-formula. Although the C-formulaonly requires the diagonalization of those Hκ matrices that cor-respond to κ points in the selected plane, the P-formula wouldrequire the diagonalization of the whole set of Hκ matrices withκ ∈ K in eq. (8).

This is also meant to demonstrate how major savings ofcomputer resources can be achieved by a suitable exploitationof crystal symmetry.



Figure 1. Scheme for the drawing of an EMD map in a region in the � planehere identified in yellow. The two shrinking factors that define the density ofthe sampling net are here set to s1 = 6, s2 = 8. See text for details.

The Problem. Consider a plane � in momentum space definedby two reciprocal lattice vectors Ba = ∑

i αaibi (a = 1, 2; i =

1, 2, 3; αai

reciprocally prime integers). Suppose we want to drawa map in a parallelogram-shaped region of the � plane definedby two vectors Zb = ∑

a zabBa and an origin P = ∑

a pa Ba (seeFig. 1). Let us define two integers s1, s2 that induce a sufficientlydense K� ≡ {κ} set in the plane (κ = ∑

a κaBa, Ba = Ba/sa, κa

integer comprised between 0 and sa − 1), and a correspondingextended grid {κ + K}, where K = ∑

a Ka Ba is one of a set Pof reciprocal lattice vectors in � such that the extended gridcovers completely the selected region. Our task is to calculateπ(p) at all points of interest of the extended grid. From thisinformation, the map is finally reconstructed using a standard2D graphical software.

Use of Symmetry. Among all point group operators of thecrystal, we can recognize the subgroup G� ≡ {V�

t } com-prising those rotation operators that leave both Ba’s in �:|B1

t B1 B2| = |B2t B1 B2| = 0. Here, Ba

t = V�t Ba and, in the

calculation of the determinant, the components of the threevectors in the {bi} basis are included columnwise. We now iden-tify within K� an irreducible set K′ and the complementaryset K′′. Any newly analyzed κ ∈ K� is assigned to K′ if it hasnot already been included in K′′. Its rotated images κ t = V�

t κ

are then compared with all κ0 ∈ K� that have not yet beenclassified. If κ0 ≈ κ t (i.e., if the two vectors coincide apart froma reciprocal lattice vector K), then κ0 is assigned to K′′, andits generating vector κ and the operator V�

t are memorized.Note that to reduce to a minimum the number of elementsin K′, a clever choice of s1, s2 may be needed (for instance,if hexagonal symmetry is present in �, they should be set toa same multiple of six). Also, within the set P , an irreduciblesubset P ′ can be identified from which all other K ∈ P vectorscan be generated through a V�

t operator.

Calculation of Eigenvectors. For each κ ∈ K′, the matrix Hκ isreconstructed and the corresponding eigenvectors Cκ and eigen-values Eκ determined [eqs. (3) and (4)]. Using the Fermi energyfrom the SCF procedure, the number nκ of occupied states isidentified (those for which ε

κ

i< ESCF

F ). For all sons of κ in K′′,the occupied eigenvectors are generated by rotation with theappropriate operator (their number is obviously the same).

Calculation of the EMD at the Extended Grid. For each κ ∈ K�,calculate P

κµν [eq. (12)]. All irreducible K′ ∈ P ′ are next considered.

For each of them,atp = κ+K′, the value ζµ = χµ(p) exp [−ıp·rµ)]is calculated for all µ’s. π(p) = ∑

µν Pκµν ζµ (ζν)

∗ is then obtained.This value is identically reproduced at all pt = V�

t p points insidethe selected region.

Examples of Application

Computational settings

All simulations to be described below are performed usingtwo periodic ab initio codes: CRYSTAL[11] and CRYSCOR.[17–19, 24] Allquantities of interest in both programs are expressed as linearcombinations of Gaussian primitives centered in high-symmetrypositions: these functions will be referred to in the followingas AOs and an explicit definition will be provided in AppendixA. The use of such a BS is mandatory for the local correlationapproach adopted in CRYSCOR.

The CRYSTAL program is here used to perform both HF andDFT calculations. Four one-electron Hamiltonians are considered:the classical HF, two typical DFT (a local density approximation(LDA)[25] and a generalized-gradient Perdew-Burke-Ernzerhof(PBE)[26]), and a hybrid Becke, three parameter, Lee-Yang-Parr(B3LYP)[27] one. The accurate calibration of the BS is perhapsthe most delicate step in defining the optimal computationalsetup when the purpose is the comparison with the experiment.As in this article, our only intention is that of analyzing the effectof some computational parameters on the computed EMD, weadopt standard, not particularly rich, BSs such as a 6-21G∗ onefor crystalline silicon and a 6-31G∗ one for lithium fluoride.

In CRYSTAL, the truncation of infinite lattice sums is controlledby five thresholds, T1–T5; the last four are here set to 8,8,8,16.The role of the first tolerance T1 will be investigated in detailin the following of this section.

The DFT exchange-correlation contribution is evaluated bynumerical integration over the cell volume: radial and angularpoints of the atomic grid are generated through Gauss–Legendre and Lebedev quadrature schemes, using a (75, 974)p

grid; grid pruning is adopted.The effect of the sampling of reciprocal space (i.e., of the

shrinking factor) will be discussed in Section Examples of 2DEMD Maps.

After completing the self-consistent calculation, CRYSTAL deter-mines via a unitary transformation of the manifold of occu-pied canonical COs, the equivalent set of Wannier functions(WFs).[28–30] WFs are real, well-localized, symmetry-adapted,mutually orthonormal, translationally equivalent functions.

WFs play an essential role in CRYSCOR, together with the com-plementary set of projected AOs (PAO); the latter are local



functions that span the virtual HF manifold and are obtainedby projecting out of each AO its “occupied” portion.[31] Thefunctions in the two sets will be concisely indicated as i, j, . . .and a, b, . . ., respectively. The MP2 energy E(2) can be writtenas a sum of all contributions Eab

ij, each corresponding to a two-

electron excitation from a pair of WFs to a pair of PAOs; therelated amplitudes are calculated via a self-consistent procedure.Exploitation of translational symmetry allows us to impose thefirst WF (i) to belong to the reference zero cell.

The input parameters of CRYSCOR serve essentially to fix threekinds of tolerances, all concerning the treatment of WFs andPAOs. The first parameter simply determines the truncation oftheir tails: in the linear combinations, which define WFs andPAOs, those AOs are disregarded whose coefficients are lowerthan tc , here set to 0.0001. The other two parameters are used toexploit the local correlation Ansatz[31, 32] according to which allexcitations can be ignored except those involving close-by WFand PAO pairs. Once the relevant WF–PAO pairs are selected, themain computational step is the evaluation of the two-electronrepulsion integrals (ERIs), (ia|jb), between the respective productdistributions. The analytical calculation of such integrals is avery demanding task; a way out of this difficulty has been toestimate the ERIs using a periodic variant of molecular density-fitting techniques,[24, 33] with extraordinary savings in computertimes and negligible loss of accuracy.

CRYSCOR and its Local-MP2 method have been applied, inthe last few years, to the study of correlation effects on theenergetic properties of many typologies of crystals like raregases,[34] ice,[35–37] TiO2,[38], adsorption of atoms and moleculeson surfaces[39, 40] and pressure-induced phase transitions.[41, 42]

In this study, the effect of the MP2 correction to the HF DMand EMD is investigated.

The effect of computational parameters

The PO Truncation Tolerance. As introduced at the end ofSection The Model: Conventions and Settings, the DM usedduring the SCF procedure in the CRYSTAL program, [Pgµν]SCF, isnot obtained according to the rigorous definition provided ineq. (8); its elements are instead truncated according to a po(po) criterion into symmetric sets of lattice vectors Gµν specificof each pair of AOs. In particular, all those elements [Pgµν]SCF aredisregarded whose corresponding po

∫χµ0(r)χνg(r) < 10−T1

where T1 is the po-tolerance (first of the five tolerances that,in CRYSTAL, control the truncation of infinite lattice sums) andχµg(r), the so-called adjoint-AO, is an s-type AO centered onposition r− rµ −g with, as exponent, the most diffuse exponentof the shell to which |µg〉 belongs.

The first computational parameter that we discuss is the potruncation tolerance T1. In Table 1, we report the number ofg-vectors that define the symmetrized sets Gµν of selected pairsof AOs of crystalline silicon as a function of T1. The selectedpairs of AOs can be grouped into three families: core–core, core–valence, and valence–virtual. The dependence of the truncationof the DM from T1 is seen to be dramatic and with a nonlinearbehavior: when passing from T1 = 8 to T1 = 15, for instance, thetruncation is the same while it starts decreasing significantly for

Table 1. Effect of the po criterion on the truncation of the [Pgµν ]SCF DM.

T1

8 15 25 35 50 100 150

core–core(2sp-2sp)1 13 13 13 19 43 135 201(2sp-2sp)2 4 4 16 28 44 104 216

core–val(2sp-3sp)1 19 19 55 87 141 429 791(2sp-3sp)2 28 28 68 80 152 456 820

val–virt(3sp-3d)1 43 43 79 135 225 627 1205(3sp-3d)2 44 44 80 140 240 664 1160

The number of g-vectors that define the symmetrized sets Gµν of selectedpairs of AOs is reported as a function of the tolerance T1 . The case of crys-talline silicon is considered with a BS (6-21G∗) made up of 1s 2sp 3sp 3d 4spAOs for each of the two atoms of silicon in the unit cell.The symbol (µ − ν)I ,with I = 1, 2 represents a pair of AOs where χµ(r) and χν(r) are centered onthe first and Ith silicon atom, respectively. The shrinking factor is si = 16 fori = 1, 2, 3, for a total of L = 16 × 16 × 16 = 4096 cells.

T1 > 15 and more rapidly for the valence–virtual than core–corepairs, as expected.

Let us stress at this point that, on one hand, the tolerance T1also governs some prescreening (based on the po criterion) ofthe two-electron repulsion integrals ERIs so that its tighteningdramatically affects the CPU time and, on the other hand, alow value of T1 (i.e., a severe truncation of the SCF-DM) givesgood results for most purposes: in the present case, for instance,passing from T1 = 8 to T1 = 100, results in a change on HFenergy and gradients of less than 1 × 10−6 % with an increaseof CPU time from ∼10 to ∼200 s.

We have seen how the truncation of the [Pgµν]SCF DM doesnot affect to a large extent the computed energies and energyderivatives; however, the question can be asked whether sucha truncation could affect or not the computation of the relatedEMD. The answer is in the positive direction; let us illustrate thiseffect in the simple case of crystalline silicon just introduced.

In the upper panel of Figure 2, we report the HF EMDπ

po

[100](p;T1) along the [100] crystallographic direction as com-puted via eq. (10) with the po-truncated [Pgµν]SCF DM for differentvalues of the tolerance T1 instead of the true P

gµν DM of eq. (8).

To magnify the effect of T1, we report in the lower panel ofthe same figure the differences between the EMDs π

po

[100](p;T1)and the “exact” reference πC

[100](p) provided by the C-formula[eq. (11)].

In both the upper and the lower panel, the two curves cor-responding to T1 = 8 and T1 = 15 are indistinguishable, asexpected from inspection of Table 1, and significantly wrongwith respect to the others and to the reference, respectively.The computed EMDs slowly converge to the exact value; how-ever, this is a very simple crystal with a small BS and this effectbecomes much more pronounced when the complexity of thesystem increases.

The AdoptedMethodological Approach. Let us discuss now, theeffect on the computed EMD of the adopted methodologicalapproach. The case of the ionic crystal of lithium fluoride (LiF)



Figure 2. Upper panel: EMD of crystalline silicon along the [100] crystallo-graphic direction,computed via the po DM [Pgµν ]SCF as a function of the adoptedtoleranceT1 (see the text for explanation); lower panel:differences between thepo-EMD and the exact one computed with the C-formula [eqs. (11) and (12)] ona very rich set of points.The adopted Hamiltonian is HF with a 6-21G∗ BS.

is considered. In Figure 3, the HF determinations of the EMDof LiF, along the three main crystallographic directions [100](upper panel), [110] (middle panel), and [111] (lower panel),are taken as a reference; the EMD differences with respect

Figure 3. EMD differences with respect to the HF ones are reported for theLiF crystal, along the three main crystallographic directions [100] (upper panel),[110] (middle panel), and [111] (lower panel). The four Hamiltonians here con-sidered are LDA, PBE, B3LYP, and HF + MP2, to be considered a representativeselection of various levels of approximation. [Color figure can be viewed in theonline issue, which is available at wileyonlinelibrary.com.]

to HF as computed with four different methods are reported.The four approaches here considered, as introduced in SectionComputational Settings, are LDA, PBE, B3LYP, and HF + MP2, tobe considered a representative selection of various levels ofapproximation.

Before discussing in detail the different features of the EMDobtained with the adopted approaches, let us introduce the for-mal connection between the EMD of the system and its kineticenergy: 〈T 〉 = 1

2

∫π(p)p2dp. Furthermore, for variational meth-

ods, like HF, the virial theorem provides the following connectionbetween the expectation values of the kinetic and potentialterms of the Hamiltonian: 〈V〉 = −2〈T 〉. As a consequence, thetotal HF energy EHF

0 of the ground state of the system can beobtained as:

EHF0 = −〈T 〉HF = −1

2

∫πHF(p) p2 dp. (13)

An index (the virial coefficient B) can be introduced to quantifythe balance between the kinetic and the potential contributionsto the total energy: B = −〈T 〉/E0. It is worth noting that thetrue energy of the system is such that B = 1. From eq. (13) itcan be inferred that, for variational methods, the informationcontained in the EMD of a system is sufficient to determine itstotal energy as well.

From inspection of Figure 3, it can be noted that the distribu-tion of electron momenta described by the three DFT methodshere considered is displaced to lower values with respect to HF,for all directions; this leads to a lower expectation value of thekinetic energy 〈T 〉, more so with LDA than PBE, and less of coursewith the hybrid B3LYP technique. This fact has been noted alsoby Thakkar with reference to molecular calculations[14, 21, 43] andis related to the failure of KS-DFT methods in satisfying thevirial theorem, which results generally in their underestimationof the kinetic energy.[7] In the present case, the virial coefficientsB are 1.000, 0.994, 0.995, and 0.996, for the HF, LDA, PBE, andB3LYP calculations, respectively.

As the virial theorem is valid for both the HF method andthe “true” system and as the HF ground state total energy isknown to lie higher than the true one (EHF > E0), we musthave 〈T 〉HF < 〈T 〉0. This means that a hypothetical experi-mental curve in Figure 3 would be negative at low momentaand positive at higher momenta; this behavior is experimen-tally observed in directional Compton profiles, which are 2Dintegration of the EMD.[7–10] This is precisely the shape of theMP2 contribution to the EMD of LiF (red lines in Figure 3) ascomputed with the CRYSCOR program. The effect of the instanta-neous electron correlation, explicitly accounted for at MP2 levelof theory, is that of increasing the global kinetic energy of thesystem with respect to the HF solution by “removing” electronsat low momenta and “adding” electrons at higher momenta.This behavior is a direct consequence of the fact that by cor-relating the electronic motions, the interelectronic repulsion isdecreased so that electrons can stay closer to the nuclei withrespect to the HF description and so go faster.



Figure 4. 2D map of (a) the total EMD and (b) the EMD-anisotropy of crystallinesilicon on the (100) plane in momentum space as computed at HF level withthe C-formula [eqs. (11) and (12)]. The sampling parameters are s1 = s2 = 8.The first Brillouin zone and its irreducible portion are reported in both panelsas black lines; all quantities are reported in a.u.

Examples of 2D EMDmaps

The EMD π(p) is a single-center function, invariant under

the symmetry operations of the point group of the crys-

tal, augmented with the inversion arising from the equality

π(p) = π(−p)[21]; such an object is a function of the counterin-

tuitive momentum-space coordinates and it is characterized by

a “collapsed” character about the origin p = 0. For these reasons,

it is generally difficult to extract the information content of the

EMD that is usually revealed in its very subtle features. In this

respect, different strategies can be followed for its analysis: (i) a

partition scheme of the total EMD of a crystal into contributions

coming from well-defined chemical subunits [π(p) = ∑i πi(p)]

has recently being proposed that takes advantage of the local-

ization of the COs into WFs[7, 28, 29, 44]; (ii) the definition of the

EMD-anisotropy �π(p) with respect to the spherical average

(SA) function πSA(|p|): �π(p) = π(p) − πSA(|p|); (iii) 2D graphi-

cal representation of EMD maps for visualizing fine features of

π(p), πi(p), and �π(p).

Total and Anisotropy EMD Maps. In Section 2D EMD Maps, we

have illustrated an efficient technique for computing 2D EMD

maps for crystals using the C-formula. We report in Figures 4a

and 4b a map of the total EMD and of the EMD-anisotropy

Figure 5. EMD differences πP(p) − πC (p) on the (100) plane of crystallinesilicon, as computed at HF level. The P-formula EMD [πP(p)] is computed withs = 4, s = 8, and s = 16 in panels (a), (b), and (c), respectively. The first Brillouinzone and its irreducible portion are reported in both panels as black lines; allquantities are reported in a.u.



of crystalline silicon on the (100) plane as computed at HF

level with this technique. The shape of the first Brillouin zone

in that plane, along with its irreducible portion, is reported

in both panels; the symmetry of both π(p) and �π(p) is

evident from these figures. The possibility of obtaining such

a representation is quite significant as both quantities can

be experimentally reconstructed from the outcomes of Comp-

ton scattering experiments.[45] The EMD reported in Figure 4

has been computed with sampling parameters s1 = s2 =8, thus corresponding to 225 κ points in the selected win-

dow (light-blue points in the figure); only 13 out of 225 are

the symmetry irreducible κ ∈ K′ points (blue points in the

figure), where the Hκ matrix is reconstructed and the cor-

responding eigenvectors Cκ and eigenvalues Eκ determined.

For the remaining 212 points, the eigenvectors are simply

rotated.[22]

The Effect of the Monkhorst Net Size on EMD Computed via the

P-Formula. In Section EMD Expressions, we have introduced

two expressions of the EMD of crystals, the P-formula of eq.

(10) and the C-formula of eqs. (11) and (12), that are equivalent

at the extended Monkhorst net points (p = κ + K), where κ

is one of the L points in the K set. We already know that at

intermediate points, the P-formula provides Fourier interpolation

values of the EMD based on the K set.

Let us investigate which is the effect of the size of K in the

computation of π(p) via the P-formula. In the case of crystalline

silicon, a cubic crystal, the number of κ points in the K set is

simply L = s3 where s is an isotropic shrinking factor.

In Figure 5, we report EMD differences πP(p)−πC(p) (where

apexes refer to the P- and C-formulae) on the (100) plane of

crystalline silicon, as computed at HF level. The reference EMD

[πC(p)] is that obtained with the C-formula and s1 = s2 = 32;

the P-formula EMD [πP(p)] is computed with shrinking factors

s = 4, s = 8, and s = 16 in panels (a), (b), and (c), respectively. In

the three panels, all the κ points corresponding to the shrinking

factor s are reported as light-blue points. By definition, those

are the points where the two expressions of the EMD, πP(p)

and πC(p), must be equal; this equality is clearly revealed in

the representation of Figure 5 where one can see how at any

point of the grid of any panel, the difference between the

two techniques is precisely zero. Between any two points in

those grids, the difference πP(p) − πC(p) can be positive or

negative or both, depending on the size of the grid: these

are the aforementioned Fourier oscillations introduced by the

P-formula. It is worth noting that when the shrinking factor s

increases, that is, when the number of κ points where the two

expressions must be equivalent increases, the Fourier oscillations

become smaller and smaller (see the scale at the right of each

panel); for instance, with s = 4 the oscillations are as large

as 0.3 a.u., they reduce to 0.03 with s = 8, and they finally

dump at 0.0006 with s = 16. At the end, we can see how the

numerical inefficiencies of the Fourier interpolation involved in

the P-formula are quite regular and a way for improving it

systematically does exist.

Conclusions

Two alternative techniques are presented for the evaluation of

the electron momentum density of crystals in the frame of

ab initio linear combination of AOs calculations with one-

electron Hamiltonians. The general formalism of the two

techniques is illustrated along with their advantages and

drawbacks.

A very efficient strategy involving the full exploitation of

both point and translational symmetry for the computation of

2D maps of EMD of crystalline materials is presented.

The effect of the most significant computational parameters

on the computed EMD is analyzed; in particular, special attention

is given to the po criterion used in the truncation of the DM

and to the subtle effect of the shrinking factor on the Fourier

relationships between the alternative expressions.

The effect of the adopted Hamiltonian on the computed

properties in momentum space is critically analyzed. The failure

of the KS-DFT in describing such quantities is discussed in terms

of its inadequacy in satisfying the virial theorem that ensures

the balance between the potential and kinetic contributions to

the total energy. A post-HF technique (MP2) for the evaluation

of DM-related properties clearly reveals the effect of electron

correlation in momentum space.

We provide in Appendix, the analytical expression of the FT

of the local basis of AOs (expressed as contractions of GTOs)

used in the formalism presented.

Appendix A: List of Symbols

In Table A1, we report a partial but hopefully useful list of the

main symbols used throughout the manuscript, along with a

short description of their meaning.

Table A1. List of symbols.

Symbol description

D Dimensionality of the system{ai} Fundamental direct lattice vectors{bi} Fundamental reciprocal lattice vectorsg General direct lattice vectorK General reciprocal lattice vector{si} Shrinking factors defining periodic boundary conditions

(PBC)L Number of direct lattice cells within PBC and, equivalently,

number of points in the Monkhorst netκ General point of the Monkhorst netK Set of L points in the Monkhorst netG Set of L cells within PBCn Number of electrons per cellr General point in direct spacep General point in reciprocal (momentum) spaceµ, ν Labels for AOsrµ , rν Positions in the cell where the AOs are centered on� One-electron DMEF Fermi energyρ(r) Electron charge densityπ(p) EMD



Appendix B: GTO-AO’s and Their FT

As it is standard practice in molecular quantum chemistry, theAOs are here contractions of GTOs of angular momentum com-ponents �, m centered in an atomic nucleus or at some selectedposition in the general cell identified by the lattice vector g:

χgµ(r − rµ) =

Mµ∑n=1

cµ,n G�,m(r − rµ − g; αµ,n), (B1)

with

G�,m(r; α) = N�,m(α)

((�)∑t

D�,mt

3∏i=1

xtii

)exp[−α r2]

and where

N�,m(α) =[

α�+ 32 22�+ 3

2 (2 − δm0)(2� + 1)(� − |m|)!π

32 (� + |m|)! (2� + 1)!!

] 12

.

Each “shell” of 2� + 1 AOs is then characterized by its centerrµ in the reference zero cell, its “type” (� = s, p, d, f , . . .), thenumber Mµ of GTOs, their “exponent” αµ,n, and their coefficientin the combination cµ,n (n = 1,Mµ). The general GTO [G�,m(r; α)]is a real normalized solid harmonics [N�,m(α) is the normaliza-tion factor] and is the product of a homogeneous polynomialof degree � in the Cartesian components of r by a Gaussianfunction with α exponent centered in the origin. The recursionprocedure for generating the polynomial coefficients D

�,mt is

described, for instance, in Ref. 22, Appendix A.The analytical expression of the FT of the general GTO is:

Fp[G�,m(r; α)] = N�,m(α) ı� (−2α)−�−1/2×

×(

(�)∑t

D�,mt

3∏i=1

ptii

)exp

[− p2

4α

].

We finally obtain:

Fp[χg

µ(r − rµ)] = exp[−ıp · (rµ + g)] Fp

[χ0

µ(r)]

≡ exp[−ıp · (rµ + g)] χµ(p); (B2)

where

χµ(p) =Mµ∑n=1

cµ,n Fp[G�,m(r; αµ,n)].

We address the reader to Ref. 46 for a detailed discussion ofthis topic.

Keywords: electron momentum density • density matrix •ab initio simulations of crystalline materials • electron correlation

How to cite this article: A. Erba, C. Pisani, J. Comput. Chem.

2012, 33, 822–831. DOI: 10.1002/jcc.22907

[1] Y. Tanaka, Y. Sakurai, A. T. Stewart, N. Shiotani, P. E. Mijnarends, S. Kaprzyk,A. Bansil, Phys.Rev.B 2001, 63, 045120.

[2] P. J. Becker, J. M. Gillet, P. Cortona, S. Ragot, Theor.Chem.Acc. 2001, 105,284.

[3] K. Nygård, M. Hakala, S. Manninen, K. Hämäläinen, M. Itou, A. Andrejczuk,Y. Sakurai, Phys.Rev.B 2006, 73, 024208.

[4] M. Hakala, S. Huotari, K. Hämäläinen, S. Manninen, P. Wernet, A. Nilsson,L. G. M. Pettersson, Phys.Rev.B 2004, 70, 125413.

[5] M. Hakala, K. Nygård, S. Manninen, L. G. M. Pettersson, K. Hämäläinen,Phys.Rev.B 2006, 73, 035432.

[6] A. Koizumi, G. Motoyama, Y. Kubo, T. Tanaka, M. Itou, Y. Sakurai, Phys.Rev. Lett. 2011, 106, 136401.

[7] C. Pisani, A. Erba, S. Casassa, M. Itou, Y. Sakurai, Phys. Rev. B 2011.DOI:10.1103/PhysRevB.00.005100.

[8] A. Erba, C. Pisani, S. Casassa, L. Maschio, M. Schütz, D. Usvyat, Phys.Rev.B 2010, 81, 165108.

[9] A. Erba, M. Itou, Y. Sakurai, R. Yamaki, M. Ito, S. Casassa, L. Maschio, A.Terentjevs, C. Pisani, Phys.Rev.B 2011, 83, 125208.

[10] C. Pisani, M. Itou, Y. Sakurai, R. Yamaki, M. Ito, A. Erba, L. Maschio, Phys.Chem.Chem.Phys. 2011, 13, 933.

[11] R. Dovesi, V. R. Saunders, C. Roetti, R. Orlando, C. M. Zicovich-Wilson,F. Pascale, K. Doll, N. M. Harrison, B. Civalleri, I. J. Bush, P. D’Arco, M.Llunell, CRYSTAL09 User’s Manual; Università di Torino: Torino, 2010.

[12] R. Dovesi, R. Orlando, B. Civalleri, C. Roetti, V. R. Saunders, C. M. Zicovich-Wilson, Z.Kristallogr. 2005, 220, 571.

[13] S. Ragot, J.Chem.Phys. 2006, 125, 014106.

[14] J. R. Hart, A. J. Thakkar, Int. J.Quantum Chem. 2005, 102, 673.

[15] J. Lethola, M. Hakala, J. Vaara, K. Hamalainen, Phys. Chem. Chem. Phys.

2011, 13, 5630.

[16] A. Erba, M. Halo, CRYSCOR09 User’s Manual; Università di Torino: Torino,2010.

[17] C. Pisani, M. Busso, G. Capecchi, S. Casassa, R. Dovesi, L. Maschio,C. Zicovich-Wilson, M. Schütz, J.Chem.Phys. 2005, 122, 094133.

[18] C. Pisani, L. Maschio, S. Casassa, M. Halo, M. Schütz, D. Usvyat, J. Comp.

Chem. 2008, 29, 2113.

[19] M. Schütz, D. Usvyat, M. Lorenz, C. Pisani, L. Maschio, S. Casassa, M. Halo,In Accurate Condensed Phase Quantum Chemistry; F. R. Manby, Ed.;CRC Press: Boca Raton, FL, 2010; p. 29.

[20] C. Pisani, R. Dovesi, A. Erba, P. Giannozzi, In Modern Charge DensityAnalysis; C. Gatti, P. Macchi, Eds.; Springer: Berlin, 2011; p. 1.

[21] A. J. Thakkar, Adv.Chem.Phys. 2004, 128, 303.

[22] C. Pisani, R. Dovesi, C. Roetti, Hartree-Fock Ab Initio Treatment ofCrystalline solids, Lecture Notes in Chemistry Series Vol. 48; SpringerVerlag: Berlin, 1988.

[23] D. Usvyat, M. Schütz, J. Phys.: Conf. Ser 2008, 117, 012027.

[24] D. Usvyat, L. Maschio, F. Manby, S. Casassa, M. Schütz, C. Pisani, Phys.Rev.B 2007, 76, 075102.

[25] J. P. Perdew, A. Zunger, Phys.Rev.B 1981, 23, 5048.

[26] J. P. Perdew, K. Burke, M. Ernzerhof, Phys.Rev. Lett. 1996, 77, 3865.

[27] A. D. Becke, J.Chem.Phys. 1993, 98, 5648.

[28] G. H. Wannier, Phys.Rev. 1937, 52, 191.

[29] C. M. Zicovich-Wilson, R. Dovesi, V. R. Saunders, J.Chem.Phys. 2001, 115,9708.

[30] S. Casassa, C. M. Zicovich-Wilson, C. Pisani, Theor. Chem.Acc. 2006, 116,726.

[31] P. Pulay, Chem.Phys. Lett. 1983, 100, 151.

[32] P. Pulay, S. Saebø, Theor.Chim.Acta 1986, 69, 357.



[33] L. Maschio, D. Usvyat, Phys.Rev.B 2008, 78, 073102.

[34] M. Halo, S. Casassa, L. Maschio, C. Pisani, Chem.Phys.Lett. 2009, 467, 294.

[35] A. Erba, S. Casassa, L. Maschio, C. Pisani, J.Phys.Chem.B 2009, 113, 2347.

[36] A. Erba, S. Casassa, R. Dovesi, L. Maschio, C. Pisani, J. Chem. Phys. 2009,130, 074505.

[37] C. Pisani, L. Maschio, S. Casassa, M. Halo, A. Erba, Theor.Chem.Acc. 2009,123, 327.

[38] D. Usvyat, L. Maschio, C. Pisani, M. Schütz, Z.Phys.Chem. 2010, 224, 441.

[39] R. Martinez-Casado, G. Mallia, D. Usvyat, L. Maschio, S. Casassa, M. Schütz,N. Harrison, J.Chem.Phys. 2011, 134, 014706.

[40] M. Halo, S. Casassa, L. Maschio, C. Pisani, R. Dovesi, D. Ehinon, I. Baraille,M. Rérat, D. Usvyat, Phys.Chem.Chem.Phys. 2011, 13, 4434.

[41] A. Erba, L. Maschio, S. Salustro, S. Casassa, J.Chem.Phys. 2011, 134, 074502.

[42] A. Erba, L. Maschio, C. Pisani, S. Casassa, Phys.Rev.B 2011, 84, 012101.

[43] A. J. Thakkar, In Theory and Applications of Computational Chemistry:The First 40 Years; C. E. Dykstra, G. Frenking, K. S. Kim, G. E. Scuseria,Eds.; Elsevier: Amsterdam, 2005; p. 11.

[44] C. M. Zicovich-Wilson, A. Erba, Theor.Chem.Acc. 2010, 126, 165.

[45] M. Cooper, Adv.Phys. 1971, 20, 643.

[46] P. Kaijser, V. H. Smith, Adv.Quantum Chem. 1977, 10, 37.

Received: 16 September 2011

Revised: 24 October 2011

Accepted: 16 November 2011

Published online on 25 January 2012


Evaluation of the electron momentum density of crystalline systems from ab initio linear combination...

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