Quantum Chemistry of Solids / 21-Basis Sets and Pseudopotentials in Periodic LCAO Calculations

rij3

8.3 Relativistic E ective Core Potentials and Valence Basis Sets

311

where c is the speed of light, V (r) is the external potential, and pˆ = −i is the momentum operator. The 4 × 4 Dirac matrices α and β in (8.36) are given by

02 σt	,			I2	02
αt = σt 02		t = (x, y, z),		β = 02	−I2		(8.37)
with a 2 × 2 Pauli spin matrix, σt		i 0		0 −1
1 0							(8.38)
σx = 0 1	,	σy = 0 −i	,	σz = 1	0

Here, 02, I2 are zero and identity 2 × 2 matrices.

The free-particle Dirac Hamiltonian provides a physical structure that the eigenvalue spectrum {Ek} consists of two parts. The states of the higher-energy spectrum, where Ek ≥ +mc2 are called the positive-energy states, and comprises states corresponding to those found in the nonrelativistic theory. The second branch of the eigenvalue spectrum consists of states with energy less than −mc2 and in a secondquantized theory they can be interpreted as states of positrons, and are called the negative-energy states.

To apply the Dirac theory to the many-particle system the one-particle Dirac operator (8.36) is augmented by the Coulomb or Coulomb–Breit operator as the two-particle term, gij , to produce the Dirac–Coulomb (DC) or Dirac–Coulomb–Breit (DCB) Hamiltonian derived from the quantum electrodynamics [496–498]:

	c	1
gij =	gij	=		, rij = ri − rj , rij = |rij |	(8.39)
			rij
	gijCB

where					(αiαj )
1			1
gijCB =		−
	rij		2		rij

+ (αirij )(αj rij ) (8.40)

By applying the independent particle approximation to many-particle relativistic DC or DCB Hamiltonians, one obtains the four-component Dirac–Hartree–Fock (DHF) method with largeand small-component spinors treated explicitly.

The DHF wavefunction Ψ is given as the Slater determinant with Ne one-particle spinors (ψi(r), i = 1, . . . , Ne), where Ne represents the number of electrons. The oneparticle spinor ψi(r) is the four-component vector whose components are the scalar wavefunctions,

	ψi2L		=	ψ1Li
ψi =				ψ2Li		(8.41)
	2S			S
	ψi			ψ3i
			ψS
				4i

The two-component electron functions ψi2L and ψi2S are called the large-component and small-component spinors, respectively, which are expanded in the basis spinors χL and χS .

The matrix DHF-LCAO equation for closed-shell systems is given as,
FC = SCE	(8.42)

312 8 Basis Sets and Pseudopotentials in Periodic LCAO Calculations

where C is a matrix of molecular spinor coe cients, E is a diagonal spinor(orbital) energy matrix, S is an overlap matrix,

Sµν

=

χµL|χνL

S0 S

(8.43)

0

χµ |χν

and the Fock matrix F is given by

Fµν =

LL + JLL

KLL

cΠLS

KLS

(8.44)

Vµν

SL

µν

−SL

µν

SS

SSµν

− SSµν

2

SS

cΠµν

− Kµν

Vµν

+ Jµν − Kµν − 2c

Sµν

¯

XX

XX

XX

¯

¯

XX

The matrices Πµν

, Vµν

, Jµν

, and

Kµν

(X, Y = L or S, L = S and S = L) are

the kinetic-energy integral, the nuclear-attraction integral and the Coulomb integral, respectively, defined by,

		¯	¯		(8.45)
	ΠµνXX = χµX |σp|χνX
	¯		¯		(8.46)
	VµνXX = χµX |V nuc|χνX
			χµ2X χν2X |χλ2Y χσ2Y
JµνXX =		PλσY Y			(8.47)
	Y =L,S λσ
and
		PλσXY χµ2X χν2X |χλ2Y χσ2Y
KµνXY =					(8.48)
	λσ
The density matrix PXY	is calculated as,
λσ
		occ
	PλσXY	= i	CiλX CiσY		(8.49)

where the negative-energy states are ignored. Even applying the four-component single-configuration (SCF) approximation, Dirac–Hartree–Fock–Breit(DHFB) or DHF methods , to calculation of heavy-atom molecules (followed by transformation of twoelectron integrals to the basis of molecular spinors) is not always an easy task because a very large set of primitive atomic basis functions can be required for such all-electron four-component SCF calculations.

In the high-Z case the e ects of relativity can be just as important as those of electron correlation, making it necessary to develop e cient methods for the simultaneous treatment of correlation and relativity [499]. Such a treatment can be made in the framework of post-DHF methods (very complicated in the practical realization) or relativistic DFT theory.

Although the DFT method has been extensively applied to nonrelativistic calculations, the four-component DFT approaches have only recently appeared (see the book [335] and the review [499] and references therein). Relativistic versions of the Kohn–Sham equations have been developed based on the relativistic extension of the Hohenberg–Kohn theory [500].

The one-electron e ective Hamiltonian for the Dirac–Kohn–Sham (DKS) method, as a relativistic extension of the conventional KS approach, takes the form,

ˆ	ˆ	δEXC [ρ, m]
hDKS = hD + VH + VXC + βσ			(8.50)
		δm(r)

8.3 Relativistic E ective Core Potentials and Valence Basis Sets				313
where hˆD is the one-electron Dirac operator (8.36), VH =	ρ(r )
		dr is the Hartree
	r r
potential, defined by the electron-charge spin density ρ(r)/. In| −contrast|			to the non-

relativistic theory the relativistic exchange-correlation potential consists of the spinindependent part (the functional derivative of the exchange-correlation energy func-

tional with respect to total density	δEXC ) and the spin-dependent part βσ δEXC		,
	δρ	δm(r)

where m(r) is the spin magnetization vector [501]. The latter is defined as m(r) =

	ψiα	and σ = (σx, σy , σz )
occ
i	ψi σψi, where ψi are two-component spinors ψi = ψiβ

is the vector of the Pauli spin matrices. In the so-called “collinear approach” the vector m(r) is projected on the z-axis to define the spin density s(r) = mS (r) =

occ	ψi σψi =	occ	ψiαψiα − ψiβ ψiβ . The total density is given by
i		i
			occ	occ
			ρ(r) = ψi ψi =	ψiαψiα + ψiβ ψiβ		(8.51)

ii

where positron states are excluded by employing the so-called no-sea approximation. The DKS equation is then the matrix pseudoeigenvalue equation by introducing basis set expansion as,

				ˆDKS	C = SCE
				h
ˆDKS	takes the form of a Fock matrix
where h
hDKS =		VµνLL + JµνLL − V(LLXC)µν					cΠµνLS
µν			cΠµνSL	VµνSS + JµνSS − V(SSXC)µν − 2c2SµνSS
In (8.53) VXX			is the exchange-correlation potential defined by,
(XC)µν
			V(XXXC)µν = χµX |		δEXC	+	δEXC
								|χνX
					δρ		δm(r)

(8.52)

(8.53)

(8.54)

During the last decade, good progress was attained in four-component techniques for molecules (see [496, 502, 503] and referencies therein) that allowed one to reduce e orts in calculation and transformation of two-electron matrix elements with small components of four-component molecular spinors. The straightforward DHF and DKS methods with the four-component spinors were implemented in several ab-initio MO LCAO programs: MOLFDIR, [504], DIRAC, [505], BERTHA, [506]. However, fully relativistic DFT calculations on molecules containing more than one or two heavy atoms are not yet routine. The e cient computational scheme for the Gaussian-based fully relativistic DHF and DKS methods is proposed in [496] and applied to hydrides MH and dimers M2 (M=Cu, Ag, Au).

As the fully relativistic (four-component) calculations demand severe computational e orts, several quasirelativistic (two-component) approximations have been proposed in which only large components are treated explicitly. The approaches with perturbative treatment of relativistic e ects [507] have also been developed in which a nonrelativistic wavefunction is used as reference. The Breit–Pauli (BP) approximation uses the perturbation theory up to the (p/mc)2 term and gives reasonable results in the first-order perturbation calculation. Unfortunately, this method cannot be used in

314 8 Basis Sets and Pseudopotentials in Periodic LCAO Calculations

variational treatment. One of the shortcomings of the BP approach is that the expansion in (p/mc)2 diverges in the case where the electronic momentum is too large, for example, for a Coulomb-like potential [496]. The zeroth-order regular approximation (ZORA), [509], can avoid this disadvantage by expanding in E/(2mc2 − V ) up to first order. The ZORA Hamiltonian is variationally stable. However, the Hamiltonian obtained by a higher-order expansion has to be treated perturbatively, similarly to the BP Hamiltonian.

Two recently developed quasirelativistic approaches are considered in [496]. In particular, in the RESC method (the relativistic scheme by eliminating small components) the Hamiltonian is separated into the spin-averaged (scalar relativistic, which can be called the one-component approximation) and spin-dependent parts. The RESC approach has several advantages. It is variationally stable. This method can easily be implemented in various nonrelativistic ab-initio programs, and the relativistic e ect is considered on the same footing with the electron-correlation e ect. RESC has been applied to various systems in ground and excited states. As the energy gradient of the RESC method is also available it is possible to study the chemical reaction in the heavy-element systems (as an example, the ionization of OsO4 is considered in [496]).

While accurate relativistic (both fourand two-component) calculations of simple heavy-atom molecules can be performed on modern computers the relativistic calculations of periodic systems are made mainly using relativistic e ective core potential (RECP). We consider these potentials in the next section.

8.3.2 Relativistic E ective Core Potentials

The two-component RECP approximation was suggested originally by Lee et al. [508] and is widely used in molecular calculations (see [487, 510, 511]). There are several reasons for using RECPs in calculations of complicated heavy-atom molecules, molecular clusters and periodic solids. As the nonrelativistic ECP approaches, the RECP approaches allow one to exclude the large number of chemically inactive electrons from calculations and treat explicitly only valence and outermost core electrons. The oscillations of the valence spinors are usually smoothed in heavy-atom cores simultaneously with exclusion of small components from the explicit treatment (quasirelativistic approximation). As a result, the number of primitive basis functions can be essentially reduced; this is especially important for calculation and transformation of two-electron integrals when studying many-atomic systems with very heavy elements including lanthanides and actinides. The RECP method is based on a welldeveloped earlier nonrelativistic technique of pseudopotential calculations; however, e ective scalar-relativistic and spin-orbit interaction e ects are taken into account by means of the RECP operator. Post-DHF correlation calculations with RECPs are performed in a natural way in the basis of spin-orbitals (and not of spinors as in all-electron four-component relativistic calculations) even for the cases when the Dirac–Coulomb–Breit(DCB) Hamiltonian is used [512]. Note, however, that the DCB technique with the separated spin-averaged and spin-dependent terms also has been developed [513], but it can be e ciently applied only in the cases when spin-dependent e ects can be neglected both for valence and for core shells. In the RECP method, the interactions with the excluded inner core shells (spinors!) are described by spindependent potentials, whereas the explicitly treated valence and outer core shells can

8.3 Relativistic E ective Core Potentials and Valence Basis Sets

315

be described by spin-orbitals. This means that some “soft” way of accounting for the core-valence orthogonality constraints is applied in the latter case [514]. Meanwhile, the strict core-valence orthogonality can be retrieved after the RECP calculation by using the restoration procedures described below. The use of the spin-orbitals allows one to reduce dramatically the expenses at the stage of correlation calculation. Thus, many complications of the DC or DCB calculations are avoided when employing RECPs.

When core electrons of a heavy atom do not play an active role, the e ective

Hamiltonian with RECP can be presented in the form
		1
HEf = [hSchr(iv ) + UEf (iv )] +			(8.55)
iv	iv >jv	riv jv

This Hamiltonian is written only for a valence subspace of electrons that are treated explicitly and denoted by indices iv and jv (large-core approximation). As in the case of nonrelativistic pseudopotentials, this subspace is often extended by inclusion of some outermost core shells for better accuracy (small-core approximation) but below we consider them as the valence shells if these outermost core and valence shells are not treated using di erent approximations. In (8.55), hSchr is the one-electron Schr¨odinger Hamiltonian

hSchr	= −	1	2 −	Zic	(8.56)
		2		r

where Zic is the charge of the nucleus decreased by the number of inner-core electrons. UEf in (8.55) is an RECP (relativistic pseudopotential) operator that is usually written in the radially local (semilocal) [510] or separable [515] approximations when the valence pseudospinors are smoothed in the heavy-atom cores. In LCAO calculations of heavy-atom molecules among the radially local RECPs, the shapeconsistent (or normconserving) RECP approaches [510] are employed and “energyconsistent” pseudopotentials by the Stuttgart–Dresden–Cologne group are also actively used [487,511,516]. The latter are now applied also in LCAO calculations of periodic systems with modified valence basis sets (see Sect. 8.3.5). To generate “energyconsistent” RECP the direct adjustment of two-component pseudopotentials is made (scalar-relativistic + spin-orbit potentials) to atomic total energy valence spectra. The latter is derived from the four-component multiconfiguration DHF all-electron atomic calculations based on the DCB Hamiltonian.The “energy-consistent” RECPs are now tabulated for all the elements of periodic table at the site www.theochem.unistuttgart.de. The adjustment of the pseudopotential parameters has been done in fully numerical atomic calculations, valence basis sets have been generated a posteriori via energy optimization. The complete set of potentials includes one-component (nonrelativistic and scalar-relativistic) e ective-core potentials (ECP), spin-orbit (SO) and core-polarization potentials (CPP); only the one-component ECPs are listed in full. The “energy-consistent”pseudopotentials are under continuous development and extension [516, 517] and the corresponding Gaussian basis sets are published [518–520].

In plane-wave calculations of solids and in molecular dynamics, the separable pseudopotentials [93,492,515] are more popular now because they provide linear scaling of computational e ort with the basis-set size in contrast to the radially local RECPs. Moreover, the nonlocal Huzinaga-type “ab-initio model potentials” [521–523] conserving the nodal structure for the valence spinors are often applied. Contrary to the

316 8 Basis Sets and Pseudopotentials in Periodic LCAO Calculations

four-component wavefunction used in DC(B) calculations, the pseudowavefunction in the RECP case can be both twoand one-component. The RECP operator simulates, in particular, interactions of the explicitly treated electrons with those that are excluded from the RECP calculations. The use of the e ective Hamiltonian (8.55) instead of all-electron four-component Hamiltonians leads to the question about its accuracy. It was shown both theoretically and in calculations, see [514], that the typical accuracy of the radially local RECPs is within 1000–3000 cm−1 for transition energies between low-lying states though otherwise is sometimes stated, [481, 482].

In a series of papers [482, 514, 524, 525] a generalized RECP(GRECP) approach was developed that involves both radially local, separable and Huzinaga-type potentials as its components. Additionally, the GRECP operator can include terms of other types, known as “self-consistent” and two-electron “term-splitting” corrections [514], which are important particularly for economical (but precise!) treatment of transition metals, lanthanides and actinides. With these terms, the accuracy provided by GRECPs can be even higher than the accuracy of the “frozen-core” approximation (employing the same number of explicitly treated electrons) because they can account for relaxation of explicitly excluded (inner core) electrons [514]. In contrast to other RECP methods, GRECP employs the idea of separating the space around a heavy atom into three regions: inner core, outer core and valence, which are first treated by employing di erent approximations for each. It allows one to attain practically any desired accuracy for compounds of lanthanides, actinides, and superheavy elements as well, while requiring moderate computational e orts since the overall accuracy is limited in practice by possibilities of correlation methods.

8.3.3 One-center Restoration of Electronic Structure in the Core Region

It should be noted that calculation of such properties as spin-dependent electronic densities near nuclei, hyperfine constants, chemical shifts, etc. with the help of the two-component pseudospinors smoothed in cores is impossible. However, the above properties (and the majority of other “core-type” properties of practical interest that are described by the operators heavily concentrated within inner cores or on nuclei) are mainly determined by the electronic densities of the valence and outer core shells near to, or on, nuclei. The valence shells can be open or easily perturbed by external fields, chemical bonding, etc., whereas outer-core shells are noticeably polarized (relaxed) in contrast to the inner-core shells. Therefore, accurate calculation of electronic structure in the valence and outer-core region is of primary interest for such properties. The electronic densities evaluated from the two-component pseudowavefunction very accurately reproduce the corresponding all-electron four-component densities in the valence and outer-core regions not only for the state used in the RECP generation but also for other states that di er by excitations of valence electrons. In the inner-core region, the pseudospinors are smoothed, so that the electronic density with the pseudowavefunction is not correct. When operators describing properties of interest are heavily concentrated near or on nuclei, their mean values are strongly a ected by the wavefunction in the inner region. The proper shapes of the valence four-component spinors must, therefore, be restored in atomic core regions after performing the RECP calculation. The applicability of the above two-step algorithm for calculation of wavefunctions of systems containing heavy atoms is a consequence of

8.3 Relativistic E ective Core Potentials and Valence Basis Sets

317

the fact that the valence and core electrons may be considered as two subsystems, interaction between which is described mainly by some integrated properties of these subsystems. The methods for consequent calculation of the valence and core parts of electronic structure give a way to combine the relative simplicity and accessibility both of RECP calculations in Gaussian basis set and of relativistic finite-di erence one-center calculations inside a sphere with the atomic core radius. In 1959, a nonrelativistic procedure of restoration of the orbitals from smoothed Phillips–Kleinman pseudo-orbitals was proposed [473] based on the orthogonalization of the latter to the original atomic core orbitals. In 1985, Pacios and Christiansen [526] suggested a modified orthogonalization scheme in the case of shape-consistent pseudospinors. At the same time, a simple procedure of “nonvariational” one-center restoration (NOCR) employing the idea of generation of equivalent basis sets in four-component DHF and two-component RECP/SCF calculations was proposed by Titov and first applied in the calculations of the PbF molecule. Later, the two-step RECP/NOCR calculations of the hyperfine structure constants and other properties were performed for the XF molecules and radicals, X=Pb,Yb,Ba [527–531], the TlF [532] and PbO [533, 534] molecules, and the molecular ion HI+, [535]. In 1994, a similar procedure was used by Bl¨ochl inside the augmentation regions [536] in solids to construct the transformation operator between pseudo-orbitals and original orbitals in his projector augmentedwave (PAW) method.

The NOCR scheme consists of the following steps [537]:

• Generation of equivalent basis sets

	fnlj (r)χljm	and smoothed two-
of one-center four-component spinors	gnlj (r)χ2j−l,jm

˜

component pseudospinors fnlj (r)χljm in finite-di erence all-electron Dirac–Fock(– Breit) and RECP/SCF calculations of the same configurations of a considered atom and its ions. The χljm is the two-component spin-angular function, the fnlj (r) and gnlj (r) are the radial parts of the large and small components of Dirac spinors, respectively. The nucleus is usually modeled by a Fermi-charge distribution within a sphere. The all-electron four-component and two-component calculations are employed to generate two equivalent numerical basis sets used at the restoration. These sets, describing mainly the atomic core region, are generated independently of the basis set used for the RECP calculations.

•The molecular pseudospinorbitals from the RECP calculation are then expanded in the basis set of one-center two-component atomic pseudospinors (for r≤Rnocr, where Rnocr is the radius of restoration that should be su ciently large for calculating core properties with the required accuracy),

Lmax j=|l+1/2|
	|		˜
˜		p		(8.57)
φp(x) ≈		cnljmfnlj (r)ωljm ,

l=0 j= l−1/2| n,m

where x denotes spatial and spin variables

•Finally, the atomic two-component pseudospinors in the basis set are replaced by equivalent four-component spinors and the expansion coe cients from (8.57) are preserved:

318 8 Basis Sets and Pseudopotentials in Periodic LCAO Calculations

Lmax j=|l+1/2|				.
	|		fnlj (r)ωljm
		p
φp(x) ≈ l=0 j= l−1/2| n,m cnljm			gnlj (r)ω2j−l,jm		(8.58)

The four-component spinors constructed in this way are orthogonal to the innercore spinors of the atom, because the atomic basis functions used in (8.58) are generated with the inner-core shells treated as “frozen”.

8.3.4 Basis Sets for Relativistic Calculations of Molecules

The fully relativistic (four-component) LCAO calculations of molecular systems use contracted Gaussian-type spinors as the basis: two scalar wavefunctions within a twocomponent basis spinor are multiplied by a common expansion coe cient, for dimensions n of both the large and small components the total number of variational parameters (the scalar expansion coe cients) is equal to 2n [496]. In the relativistic correlated calculations the atomic basis sets should be optimized in the atomic correlated calculations. As Alml¨of and Taylor showed [538], atomic basis sets optimized to describe correlations in atoms also describe correlation e ects in molecules very well. The two main types of basis sets are used in correlation calculations of molecules: basis of atomic natural orbitals (ANO) suggested by Alml¨of and Taylor [538]; correlationconsistent (CC) basis set suggested by Dunning [462].

Atomic natural orbitals {ψp} are obtained by unitary transformation of some set of orthonormal basis functions {φp}. Natural orbitals diagonalize density matrix Dij for some state (or group of the states) of the atom or its ions:

ρ(r, r ) = Dpqφp†(r)φq(r ) =	npψp†(r)ψp(r )	(8.59)
p,q	p

where np are occupation numbers. Only those {ψp} are selected in the ANO basis set for which np is greater than some threshold: np ≥ nthr; nthr 10−3 − 10−5.

Unlike the ANO scheme, the di erent number of primitive Gaussian functions are used in each basis set in the CC approach. Exponents of the polarization-correlation functions are optimized in correlation calculations of the atomic terms to provide the lowest possible total energy. It turns out that basis functions can be separated in groups and every function in each group lowers the total energy approximately by the same amount. Correlation-consistency means that whole groups of functions are added to the basis set, not just separate functions.

Both ANO and CC schemes have a number of advantages, i.e. the ANO basis set is relatively easy to construct and the smaller ANO basis is just a subset of the larger one. Advantages of the CC basis are the smaller number of the primitive functions and a natural criterion for estimation of the completeness of the basis set. The following disadvantages can be emphasized for CC and ANO basis sets:

•for heavy atoms with many electrons, the number of ANO and CC functions becomes too large to obtain satisfactory accuracy in transition energies and prop-

erties. Even for the light Ne atom, ANO basis sets obtained in correlation CI-SD calculation are [3s 2p 1d] with nthr 10−3 and [5s 4p 3d 2f 1g] for nthr 10−5;

•“occupation number-optimized” (ANO) or “energy-optimized” (CC) basis sets do not take into account special features of one or another property and do not reflect properly the state of the “atom-in-molecule”;

8.3 Relativistic E ective Core Potentials and Valence Basis Sets

319

•the incompleteness of atomic basis sets with the change of internuclear distance leads to basis-set superposition error (BSSE). This error is usually corrected by a “counterpoise” correction [539] that is calculated rather arbitrarily (it depends, in particular, on the atomic state used for its calculation) for ANO and CC basis sets.

The generalization of the correlation-consistent scheme of the basis-set generation was suggested, see [524] and references therein, which allows one to control e ectively the quality of the basis set in di erent space regions depending on the property of interest. Practically all properties can be divided into two groups: (1) “valence” properties, which are determined by the wavefunction in the valence region, like dissociation energy or transition energies and (2) “core” properties, which are described by operators heavily concentrated near the nuclei of the heavy atom, like the hyperfine structure. The calculations of the di erent kind of properties require, in principle, di erent optimization criteria for basis sets. The generalized correlation (GC) basis set is constructed in the following stages:

1 Generation of the trial spin-orbitals, two-component spinors or four-component spinors in the atomic SCF calculation with numerical functions. Functions are obtained in SCF calculations of the atom and its ions, thus trial functions are localized in valence or outer-core regions.

2Criterion of choice of the optimal function is the maximization of some functional that depends on total energies of the states and transition energies between them. The energies of the states are obtained in some correlation calculation of the group of the states that describe an “atom-in-molecule” in the best way. The choice of

the energetic functional is the key element of the GC-basis generation.

3The resulting sets of functions are approximated by Gaussian functions with the same exponents for each l (or pair of lj) by the least square fit.

A commonly used choice of the functional is:

		1	M
	Fc(∆E1, . . . , ∆EM ) =		i	∆Ei	(8.60)
		M

Here, Fc is the change in the average energy for some considered group of the lowestlying terms.

The functional used in the GC generation scheme can be chosen as follows:

M

Fv(∆E1, . . . , ∆EM ) = max |∆Ei − ∆Ej | (8.61)

i>j

where M is the number of atomic terms, ∆Ei ≥ 0 is the lowering of the total energy of the ith term in comparison to calculation without the given function. Thus, Fv presents the maximal change in transition energies. (Some other versions of the functional on changes of transition energies can also be used but their results are not very di erent.)

Functional Fc is reasonable to use for generation of the GC basis for “core” properties, while Fv can be used when one is interested in “valence” properties. When only valence electrons are correlated, any functional can be used since the function giving the maximal change in total energy, as a rule, also provides the maximal change in

320 8 Basis Sets and Pseudopotentials in Periodic LCAO Calculations

transition energies in this case. In a universal scheme, both of the above functionals can be applied together when Fc is used in the first step to select the most important correlation functions and then, Fv is used (see also [540] for a variational justification of the suggested functionals). If the e ect of atomic polarization is important the GC basis can be augmented by di use polarization functions, whose exponents are optimized in the SCF calculation of an atom in an electric field. The valence basis sets, used in relativistic molecular calculations, have to be in correspondence with RECP used in the basis set optimization. As was already mentioned the basis sets corresponding to the “energy-consistent” RECP can be found on the site www.theochem.uni-stuttgart.de. Highly accurate relativistic Gaussian basis sets are developed for the 103 elements from H to Lr [544]. Orbital exponents are optimized by minimizing the atomic selfconsistent field (SCF) energy with the scalar relativistic approximation. The basis sets are designed to have equal quality and to be appropriate for the incorporation of relativistic e ects. The basis-set performance was tested by calculations on prototypical molecules, hydrides, and dimers of copper, silver, and gold using SCF, MP theory, and the single and double coupled-cluster methods. Spectroscopic constants and dissociation energies are reported for the ground state of hydrides, which agree well with the experimental data (the mean absolute error relative to the experiment in dissociation energy, equilibrium bond length and harmonic frequency is 0.09eV , 0.003 ˚A, and 2 cm−1, respectively). The optimized Slater-type basis sets for elements 1–118 are developed in [545]. The exponents of the Slater-type functions are optimized for the use in scalar-relativistic zero-order regular approximation (ZORA). These basis sets are used in DFT code ADF [345] for calculations of molecules and solids. The use of RECP and valence basis sets for periodic systems is considered in the next section.

8.3.5 Relativistic LCAO Methods for Periodic Systems

The influence of relativistic e ects on the structural and electronic properties of the crystals with heavy elements is well known [541].

One important e ect is the strong relativistic contraction of s and p orbitals due to the non-negligible probability to find them close to the nucleus; in turn d and f orbitals are stretched due to indirect relativistic e ects. This largely a ects the interatomic distances, the cohesive energies, the one-electron energies, etc. This e ect can be partly handled within the scalar relativistic (SR) calculation, neglecting spin-orbit interaction. Another important relativistic e ect is due to the spin-orbit coupling, which plays an essential role in the magnetic properties of solids [546]. The spin-orbit coupling requires more elaborate treatment beyond the SR calculations, i.e. the twoor four-component relativistic calculations.

The four-component DHF LCAO equations for 1D-, 2Dand 3D-periodic systems were at first presented by Ladik [547]. The resulting somewhat complicated generalized matrix eigenvalue equation for solids is described (for details we refer the reader to [547]). It was also shown for 1D and 2D systems how MP2 methods could be applied in their relativistic form. With the help of these, on the one hand, the total energy per unit cell (including correlation e ects) can be computed. On the other hand, the relativistic band structure can also be corrected for correlation. Note that the symmetry of crystalline orbitals changes, compared with the nonrelativistic case, as the symmetry of the DHF Hamiltonian is described by double space groups. Finally,