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

	8.2	Nonrelativistic E ective Core Potentials and Valence Basis Sets				301
Vˆ P S f	r,θ,ϕ	=	lm Ylm(θ, ϕ)Vl(r)		d(cos θ )dϕ Ylm(θ , ϕ )f (r, θ , ϕ )	(8.15)

All the PP information is in the radial functions Vl(r). (We note that the HF exchange operator is fully nonlocal both in the angular and in the radial variables.) To generate PP an all-electron calculation (HF or DFT) of a free atom is performed. The DFT PP Hamiltonian includes the local (Hartree and exchange-correlation) and semilocal (PP) parts; the HF PP Hamiltonian includes local (Hartree), nonlocal (exchange) and semilocal (PP) parts. The set of PP parameters is chosen to accurately reproduce the eigenvalues and eigenfunctions of valence states. Clearly, in the region of space in which most of the electronic norm is concentrated both orbitals must be very close, if not identical. On the other hand, the form of the valence orbitals in the core region, where the core electrons are moving, is less relevant. Otherwise, the core states themselves would play a more important role. In the core region one can thus allow the pseudo-orbital (PO) to di er from the all-electron orbital (AO) without losing too much accuracy [470].

The calculations of real systems (for example, color centers in ionic crystals, [475]) were made based on the model potential of Abarenkov and Heine [476]:

		Vval(r)					Al(E)
VM =			Z		− lm |			|	(8.16)
	−		r	+ Vval(r),					r > rc

where Vval is the Coulomb and exchange potential due to the valence electrons, Z is the valence charge, and Al(E) is a constant (in space) function of energy E chosen to make the PO logarithmic derivative equal that of the true eigenfunction at several atomic eigenvalues. To a high degree of accuracy Al(E) may be linearized: Al(E) = Al +BlE.

A giant step forward in pseudopotentials was taken by Hamann et al. [472], who introduced norm conserving pseudopotentials (NCPP). NCCP for angular momentum l is chosen so that

(1) the resulting atomic valence PO agrees with the corresponding all-electron (AE) AO for all r larger than some l-dependent cuto (core) radius

	RnlP S (r) = RnlAE (r)		(8.17)
(2) the norm of the orbital is conserved,
rc,l	rc,l
0	|RnlP S (r)|2dr = 0	|Rnlae(r)|2dr	(8.18)

(3) The logarithmic derivatives of the true and pseudowavefunctions and their first energy derivatives agree for r > rc.

Condition 1 automatically implies that the real and pseudovalence eigenvalues agree for a chosen “prototype” configuration, as the eigenvalue determines the asymptotic decay of the orbitals.

Properties (2) and (3) are crucial for the pseudopotential to have optimum transferability among a variety of chemical environments [470].

The PP concept has been motivated by the inertness of the atomic core states in binding so that the ionic core of the atom provides a fixed potential in which the

302 8 Basis Sets and Pseudopotentials in Periodic LCAO Calculations

valence electrons are moving, independently of the system (atom, molecule or solid) that is considered. However, in polyatomic systems the valence states undergo obvious modifications compared to the atomic valence orbitals, even if the polyatomic core potential is given by a simple linear superposition of atomic core potentials. Most notably, the eigenenergies change when packing atoms together, which leads to bonding and antibonding states in molecules and to energy bands in solids. Thus, while PPs are designed to reproduce the valence AOs of some chosen atomic reference configuration (usually the ground state), it is not clear a priori that they will have the same property for all kinds of polyatomic systems and for other atomic configurations. Consequently, one has to make sure that the PP is transferable from its atomic reference state to the actually interesting environment. To check the transferability of PP one has to analyze the sensitivity of the agreement between atomic POs and AOs to the specific eigenenergy in the single-particle equation. One finds that the variation of the logarithmic derivative RnlP S (r)/Rnlae(r) with the single-particle energy is determined by the norm contained in the sphere between the origin and r. This is true in particular in the neighborhood of one of the actual atomic eigenvalues εnl, i.e. for a bound atomic eigenstate. Thus, as soon as normconservation is ensured, the POs exactly reproduce the energy dependence of the logarithmic derivative of the AOs for r > rc,l. Consequently, one expects the POs to react as the AOs when the valence states experience some energy shift in a polyatomic environment – provided the underlying PPs are normconserving. This argument supporting the transferability of PPs emphasizes the importance of normconservation in a very explicit way. In practice, it is, nevertheless, always recommended to check the transferability explicitly by examination of some suitable atomic excitation process and of the binding properties of simple molecular or crystalline systems [470]. In the next section we consider pseudopotentials used in modern periodic calculations.

8.2.2 Gaussian Form of E ective Core Potentials

and Valence Basis Sets in Periodic LCAO Calculations

The form of ECP used in the condensed-matter applications depends on the basis chosen – PW or LCAO. The numeric pseudopotentials in plane-wave calculations must be used with the density functional that was employed to generate them from a reference atomic state. This is a natural and logical choice whenever one of the planewave DFT codes is used. In PW calculations the valence functions are expanded in Fourier components, and the cost of the calculation scales as a power of the number of Fourier components needed in the calculation. One goal of PP is to create POs that are as smooth as possible and yet are accurate. In PW calculations maximizing smoothness is to minimize the range of Fourier space needed to describe the valence properties to a given accuracy [10]. Normconserving PPs achieve the goal of accuracy, usually at some sacrifice of smoothness. A di erent approach by Vanderbilt, known as “ultrasoft pseudopotentials”(US) [477] reaches the goal of accurate PW calculations by a transformation that re-expresses the problem in terms of a smooth function and an auxiliary function around each core that represents the rapidly varying part of the density. The generation code for Vanderbilt US pseudopotentials and their library can be found on site http://www.physics.rutgers.edu/ dhv/uspp.

Ab-initio pseudopotentials for PW calculations of solids can be generated also by the fhiPP package [478], see also site http://www.fhi-berlin.mpg.de/th/fhimd/.

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The numerical AO-based DFT code SIESTA [344] employs the same numeric pseudopotentials as plane-wave-based codes. An alternative approach is used in the Slater-orbital-based DFT code ADF [345], where so-called core functions are introduced. They represent the core-electron charge distribution, but are not variational degrees of freedom and serve as fixed core charges that generate the potential experienced by valence electrons [479].

In molecular quantum chemistry Gaussian-function-based computations, e ective core potentials were originally derived from a reference calculation of a single atom within the nonrelativistic Hartree–Fock or relativistic Dirac–Fock (see Sect. 8.3) approximations, or from some method including electron correlations (CI, for instance). A review of these methods, as well as a general theory of ECPs is provided in [480,481].

In this section we discuss those e ective core potentials and the corresponding valence basis sets that are used for Gaussian-function-based LCAO periodic computations implemented in the computer codes CRYSTAL [23] and GAUSSIAN [107].

The ECP general form is a sum of a Coulomb term, a local term and a semilocal term

						M
VP S (r) = C + Vloc + Vsl = −		ZN
			r		+	rnk Ck exp(−αkr2)
	l					k=1
						rnkl −2Ckl exp(−αklr2)# Pˆl
+	3		" Ml				(8.19)
	=0			k=1

where ZN in a Coulomb term is the e ective nuclear charge (total nuclear charge minus the number of electrons represented by ECP). The local term is a sum of products of polynomial and Gaussian radial functions. A semilocal term is a sum of products of polynomial radial functions, Gaussian radial functions and angular-

ˆ

momentum projection operators Pl. Therefore, to specify semilocal ECP one needs to include a collection of triplets (coe cient, power of r and exponent) for each term in each angular momentum of ECP.

Hay and Wadt (HW) ECP [483] are of the general form (8.19). The procedure employed for generation of ECPs includes the following sequence of steps: 1) the “core” orbitals to be replaced and the remaining “valence” orbitals are defined. This step defines whether the small-core (the outermost core electrons are explicitly treated along with the valence electrons) or a large-core HW pseudopotential is generated; 2) the true numerical valence orbitals are obtained from self-consistent nonrelativistic Hartree–Fock (or relativistic Dirac–Fock) calculations for l = 0, 1, . . . , L, where L, in general, is one greater than the highest angular-momentum quantum number of any core orbital; 3) smooth, nodeless pseudo-orbitals (PO) are derived from the true Hartree–Fock (Dirac–Fock) orbitals in a manner so that PO behave as closely as possible to HF orbitals in the outer, valence region of the atom; 4) numerical e ective core potentials VlP S are derived for each l by demanding that PO is a solution in the field of Vl with the same orbital energy ε as the Hartree–Fock (Dirac–Fock) orbital; 5) the numerical potentials are fit in analytic form with Gaussian functions, the total potential is represented as (8.19); 6) the numerical POs are also fit with Gaussian functions to obtain basis sets for molecular or periodic calculations. In the case of large-core ECP the primitive Gaussian bases (3s2p5d), (3s3p4d) and (3s3p3d) are tabulated for the first, second, and third transition series atoms, respectively. The

304 8 Basis Sets and Pseudopotentials in Periodic LCAO Calculations

figures in brackets mean the number of primitive of Gaussians in ns, np and (n − 1)d contracted AOs for n = 4, 5, 6. In the case of small-core ECP (n − 1)s, (n − 1)p contracted AOs are added and given as the linear combinations of primitive Gaussians. Hay-Wadt ECPs and valence-electron basis sets are also generated for main-group elements: large core – for Na to Xe, and Cs to Bi, small core – for K, Ca, Rb, Sr, Cs, Ba.

The other known ECP and valence-electron basis sets were generated using the procedure, described for Hay–Wadt ECP generation. Durand–Barthelat large-core semilocal ECP [484] and corresponding valence-electron basis sets are generated for 3d-transition elements and the main-group elements Li to Kr.

Compact oneand two-Gaussian expansions for the components of the e ective potentials of atoms in the first two rows are presented by Stevens–Basch–Krauss [485]. Later, the list of ECP was extended to the third-, fourthand fifth-row atoms [486] and includes relativistic ECP (RECP). The pseudo-orbital basis-set expansions for the first two rows of atoms consist of four Gaussian primitives using a common set of exponents for the s and p functions. Analytic SBK RECP are generated in order to reproduce POs and eigenvalues as closely as possible. The semilocal SBK ECP are

given by
	r2Vl(r) = Alkrnl,k exp(−Bl,kr2)	(8.20)

k

The potentials and basis sets have been used to calculate the equilibrium structures and spectroscopic properties of several molecules. The results compare extremely favorably with the corresponding all-electron calculations.

Stuttgart–Dresden (SD)ECP (formerly Stoll and Preuss ECP) are under constant

development [487]. SD semilocal ECPs are written in the form
Vsl =	3	" µk	rnkl −2Ckl exp(−αkr2)# Pˆl	(8.21)
	=0	k=1
	l

Note the di erent convention for the factor rnkl −2 compared to (8.19). The database of SD ECP include relativistic ECP (RECP) generated by solving the relativistic Dirac–Fock equation for atoms. Improved SD pseudopotentials exist for many of the main-group elements, and the pseudopotentials are also available for 5d and other heavier elements. The most recent ECP parameters, optimized valence-electron basis sets, a list of references and guidelines for the choice of the pseudopotentials can be found at site http://www.theochem.uni-stuttgart.de. SD ECP can be used in periodic LCAO CRYSTAL and GAUSSIAN codes [23, 107].

Pseudopotentials are also used as embedding potentials when some special region of a covalently bonded solid or very large molecule is modeled by a modest-size cluster. The embedding pseudopotentials are considered in the next section.

8.2.3 Separable Embedding Potential

As was noted in the preceding section the use of atomic pseudopotentials (or e ective core potentials–ECPs) considerably simplifies the quantum-mechanical description of polyatomic systems (molecules and crystals) as the much more localized and chemically inert core electrons are simulated by ECP introduction. The choice of the norm

8.2 Nonrelativistic E ective Core Potentials and Valence Basis Sets

305

conserving and transferable ECPs ensures that the valence states are reproduced in the majority of cases as accurately as would be done in all-electron calculations.

The second important application of pseudopotentials is connected with the socalled embedded-cluster model. This model is applied when one is interested in the electronic structure and properties of some small region of a large system such as a localized point defect in a solid, an adsorbed molecule on a solid surface or an active site in a very large biological molecule. In such a case one may model the region of interest by cutting a modest-sized but finite cluster out of the larger system and performing the calculation on it. For ionic solids the surrounding crystal can be modeled by the system of point charges. More di cult is the case when the environment is a covalently bonded system and the boundary between the cluster and environment passes through chemical bonds (covalent or partly covalent). In this case, the distant and nearest parts of an environment should be treated separately [488]. In the distant part, only the electrostatic potential representing the ionic component of an environment should be retained. The nearest part, corresponding to the “broken or dangling bonds”, needs special consideration. Each atom of the cluster boundary surface has unsaturated “dangling bonds” that cause spurious e ects unless saturated in some way, usually by adding a hydrogen or pseudoatoms there [489]. This is better than leaving the dangling bond but clearly the termination is still imperfect in the sense that the bond to the attached atom is di erent from the bond to whatever atom is situated there in the real system [488]. Use of the embedding potentials (EP) to saturate dangling bonds of the cluster gives a cluster surface bond identical to that in the real large system.

Recently introduced new separable potentials [488, 490] have several kinds of applications: 1) when some special region of a covalently bonded solid or very large molecule is modeled by a modest-sized cluster, each dangling bond at the cluster surface can be saturated in a way that exactly reproduces the bond in the complete system; 2) a similar approach can be used at the matching surface in an embedding scheme for calculations on the same type of systems; 3) application to atomic e ective core potentials where the new potential operator avoids the possibility of “ghost” states that sometimes plague the widely used pseudopotentials.

The important property of the introduced embedding potential is its separability.

ˆ

Let the nonlocal operator V (r) depend on the space variable r only and be represented

ˆ

by the integral operator V :

Vˆ ψ(r) = v(r, r )ψ(r )dr

(8.22)

with the kernel v(r, r ). Expanding the kernel in both arguments with the complete orthonormal set of functions ϕi(r)(i = 1, 2, 3, . . .) the following representation of the potential can be obtained:

	∞
ˆ		(8.23)
V =	|ϕi Vij ϕj |

i,j=1

where Vij is the infinite Hermitian matrix. If this matrix has a finite rank k, then only k of its eigenvalues are not equal to zero and the corresponding potential has the form

	k
ˆ	i	(8.24)
Vsep =	|χi µi χi|
	=1

306 8 Basis Sets and Pseudopotentials in Periodic LCAO Calculations

where χi(r) and µi are eigenfunctions and eigenvalues of the potential. Equation (8.24) defines the separable potential as a finite sum of single-state projectors (|χi χi|) with constant weights µi. A separable operator is an operator with a finite-dimensional functional space for its range. If one employs another orthonormal basis set ϕi(i = 1, 2, . . . , k) that spans the same functional space as the set χi(r)(i = 1, 2, . . . , k), the separable potential will have the nondiagonal form

	k
ˆ		(8.25)
Vsep =	|ϕi Vij ϕj |

i,j=1

but with finite sum over i, j. The di erence between the separable potential and the potential of the general form is that the separable potential with a suitable choice of the basis could be transformed to the form (8.24) or (8.25) and the potential of the general form could not.

As an example of separable potentials we mention the semilocal ECP, see (8.14), which have a form similar to (8.24), but where µi = Vi(r) are functions of the radial variable. These semilocal potentials are separable only in the angular variables (in which these potentials are nonlocal) as the separability is the property of nonlocal potentials.

Separable nonlocal ECPs were extensively studied [474, 491–494]. The nonlocal separable potentials application is complicated by the problem of “ghost states” [495] i.e. extra bound states with levels, under the reference atomic eigenenergy. For semilocal ECPs, used in modern computer LCAO codes, this problem does not occur [493], for the embedding potential this problem has to be taken into account.

ˆ 0 0

Let us consider a Hamiltonian H0 with eigenfunctions Ψi and eigenvalues Ei . Suppose we have an arbitrary set of orthonormal functions Ψp and an arbitrary set of real numbers Ep(p = 1, . . . , n). The aim is to develop another Hamiltonian

		ˆ	ˆ	ˆ	(8.26)
		Hsep = H0		+ Vsep
	ˆ				ˆ	are
with separable potential Vsep		so that n eigenfunctions and eigenvalues of Hsep
				ˆ	are equal to
exactly Ψp and Ep, whereas all other eigenvalues Ei, i > n of Hsep
	ˆ
eigenvalues of H0.			ˆ		0
ˆ	Of course, eigenfunctions Ψi, (i > n) of Hsep will di er from eigenfunctions Ψi					of
					0
H0	because the former are orthogonal to Ψp, (p ≤ n) and the latter to Ψp .
	ˆ	ˆ
	The transformation H0 into Hsep could be done in two steps.

ˆ ˆ ˆ ˆ †

In the first step the similarity transformation H = U H0U with a unitary operator

ˆ

U is made, which changes the eigenfunctions and leaves eigenvalues unchanged. To find the desired similarity transformation the following auxiliary problem is considered: a

ˆ				0	into
unitary operator U is found that transforms one set of orthonormal functions Ψp
another set of orthonormal functions Ψp:
ˆ	0	|αp	| = 1, p = 1, 2, . . . , n	(8.27)
U Ψp = αpΨp,

and there is no linear dependence among functions Ψp0 and Ψp, i.e. the dimension of functional space spanned by both sets of functions together is equal to 2n. The

ˆ	ˆ ˆ	ˆ	ˆ	n	|χp Tpq χq| is a separable
operator U is taken in the form U = I		− R and R = p,q=1

8.2 Nonrelativistic E ective Core Potentials and Valence Basis Sets

307

operator with the smallest possible dimension. Dropping the mathematical details

						ˆ
given in [488], we write the explicit form of the operator R:
n		(		3
	0		−1		0
ˆ		− αpΨp Q				− αqΨq|	(8.28)
R =	|Ψp			pq	Ψq

p,q=1

where χp = Ψp0 − αpΨp and Qpq = χp|Ψq0 = δpq − αp Ψp|Ψq0 . The second step is the energy-shift transformation that changes specified eigenvalues only:

						n
			ˆ	ˆ
			Hsep = H			+	|Ψp (Ep −			Ep ) Ψp|
						p=1
						n
			ˆ ˆ	ˆ †
			= U H0U			+	|Ψp (Ep −			Ep ) Ψp|	(8.29)
						p=1
		ˆ
Hence the potential Vsep can be written as
										n
ˆ	ˆ	ˆ	ˆ ˆ ˆ	†		ˆ ˆ	ˆ ˆ	†	+		(8.30)
										0
Vsep = Hsep − H0			= RH0R		− RH0		− H0R			|Ψp (Ep − Ep ) Ψp|

p=1

This potential is separable with dimension n, i.e. can be represented in the form

	3n
ˆ		(8.31)
Vsep =	|fi Vij fj |

i,j

ˆ ˆ ˆ ˆ† ˆ ˆ ˆ† where the expression for Vij can be found from expressions of RH0, H0R and RH0R .

Although all equations were developed for arbitrary phase factors αp, in applications the real orbitals are used. For the sake of simplicity all the phase factors αp can be selected to be real, which makes the separable potential real as well. The signs of Ψp

0	0			ˆ		0 when
and Ψp	0are selected so that Ψp|Ψp ≥ 0 and all αp = −	1 to make V			sep →
			0
G	(G is a collection of ΨP , Ep, p = 1, 2, . . . , and G			is a similar collection of
G0→ 0
Ψp , Ep ).

The separable embedding potential (8.31) was applied to model the single chemical bond between the atom A of the cluster and the atom B of the cluster environment [488]. To simulate the e ect of the cluster environment the atom B is replaced by a pseudoatom BP S at the same position as the actual atom B. The influence of the pseudoatom on the cluster is described by the potential that was assumed to have the

following form:		ˆ	ˆ	ˆ
	ˆ				(8.32)
	Vpsat = VH + Vsl + Vsep
ˆ	= −1/r is the hydrogen potential (in a.u.), responsible for the asymptotic
Here, VH
	ˆ
behavior of the potential, Vsl is a semilocal potential
	ˆ
	Vsl =	Vl(r)|Ylm Ylm|			(8.33)
		lm
ˆ					ˆ
and Vsep	is a case n = 1 separable potential (8.31) defined by the Hamiltonian H0,

the energy E10 and the wavefunction Ψ10 of this Hamiltonian ground state, and by the

308 8 Basis Sets and Pseudopotentials in Periodic LCAO Calculations

					ˆ	ˆ	ˆ	was taken the
energy E1 and the wavefunction Ψ1 of the operator H0						+ Vsl. As H0
following		1
	ˆ			ˆ	ˆ
	H0	= −	2	∆ + VH + Vsl				(8.34)
ˆ	is selected so as to reproduce the spectrum of the excited states of the
where Vsl

atom B similarly to what is usually done in the calculation of the e ective atomic core potentials. The wavefunction Ψ10 of the ground state of this Hamiltonian is a spherically symmetrical 1s function and it is not at all a directed hybrid orbital of the atom B that makes the single bond with the atom A. The separable potential

ˆ	ˆ	0	for function Ψ1
Vsep being added to H0		will change the ground-state wavefunction Ψ1

without changing the excited-state spectrum. Therefore, Ψ1 should be considered as a hybrid orbital of the atom B that is not known. However, the bond orbital ϕbond can be found. The new localization criterion was proposed to calculate the localized on bond A − B orbital ϕbond. This criterion is based on the maximization of the singleorbital contribution to the Wiberg index WAB defined by (4.137). The one-electron density matrix is calculated in the bases of AOs orthogonalized according to L¨owdin procedure [225]. This bond orbital depends on Ψ1 implicitly and in [488] a method to calculate ϕbond and its e ective energy Ebond was developed.

The transferability of the separable embedding potential used seems very likely as the developed method of pseudopotential calculation results in the one-center poten-

ˆ

tial (all the components of this potential, including Vsep, are centered on the atom B). This is demonstrated by use of the proposed separable potential (8.32) for the pseudosilicon atom that is to substitute the −SiH3 radical in X–SiH3 molecules (X was taken to be H, F, Cl, Br, and I). In all these molecules the hydrogen or halogen constitute the single chemical bond with Si. This bond is broken when the radical −SiH3 is removed and it could be saturated with the pseudosilicon atom so that the original five-atom molecule would be replaced by the diatomic molecule X–Sips. The angles between bonds in SiH3X molecules vary only a little and the perfect tetrahedron angle was used for all molecules. The Si–H bond length in these molecules is also approximately equal to that in the SiH4 molecule, and silicon–halogen bond length varies considerably (the experimental values for silicon–halogen bond lengths were taken from diatomic Si–Hal molecules). The potential (8.32) was generated for the pseudosilicon atom to make the bond in the two-electron diatomic molecule H–Sips the same as the single chemical bond H–Si in the SiH4 molecule, this potential is centered at the Si atom.

The potential-generation procedure consists of several stages. In the first stage the all-electron HF calculations on the SiH4 molecule were performed. The canonical HF orbitals, orbital energies, and the total density matrix were obtained in the GTO basis. In the second stage, the noncanonical HF orbital ϕbond localized on the selected Si-H bond was calculated using the Wiberg index for the localization criteria. The maximum localizing functional value obtained was 0.965, which is close enough to the maximum possible value 1 for the single covalent bond. In the third stage, the

ˆ

semilocal potential Vsl was generated so that the excited energy levels of the equation

	1	1			ˆ	0	0	0
−		∆ −			+ Vsl	ψnlm	(r) = Enl	ψnlm	(8.35)
	2		r

8.2 Nonrelativistic E ective Core Potentials and Valence Basis Sets

309

coincide with unoccupied energy levels of the neutral silicon atom (E20s =E(Si)4s, E20p = E(Si)4p, E30d = E(Si)3d and so on). No adjustment was done for the E10s energy level and it could be quite di erent from E(Si)3s. In the fourth and final stage, the function Ψ1 and the energy E1 were determined, and the pseudosilicon potential was obtained. The obtained potential of the pseudosilicon atom was used to calculate the diatomic molecules X−Sips(X = F, Cl, Br, I). The properties of the bond and the state of the halogen atom in the diatomic molecule could be compared with those in the real five-atom molecule. This comparison will show whether the same pseudosilicon potential could be used in all considered molecules, i.e., whether the generated pseudosilicon potential is transferable. The dipole moment of the bond was chosen for the bond property, and the atomic charge for the halogen property. For the diatomic molecule the dipole moment of the bond is the dipole moment of the molecule itself. However, for the five-atom molecule the dipole moment of the particular bond is not defined a priori. To make the comparison possible several assumptions were used that are natural enough. It was assumed that the dipole moment of every fiveatom molecule SiH3X is the sum of the dipole moment of the X–Si bond and the dipole moment of the SiH3 radical. Next, it was assumed that the dipole moment of the (SiH3) radical is the same in all five molecules. The latter assumption can be considered as the “frozen bond” approximation, i.e. the neglect of the radical polarization when the X–Si bond is changing. Other details of the dipole-moment calculations can be found in [488]. To improve the agreement of the calculated dipole moments with the experimental data one d-orbital was added to the GTO 6-311G basis of silicon and halogen atoms.

In Tables 8.4 and 8.5 the dipole moments and Mulliken atomic charges found in the all-electron HF calculations are compared with those obtained with the use of a pseudosilicon atom and a hydrogen atom to saturate the bond. The disagreement between experimental and HF data is explained by the neglect of the correlation e ects, and in the case of a SiH3I molecule the neglect of relativistic e ects. The analysis of Table 8.4 allows us to conclude that the pseudosilicon potential generated for one molecule (SiH4) to represent the SiH3 radical could be employed in calculations of other molecules (SiH3X, X = F, Cl, Br, I) with the same radical.

From Table 8.5 it follows that the correlation between charges calculated with di erent methods is similar to that for the dipole moments. However, it should be remembered that atomic charges are not physically observable quantities and they strongly depend on their definition and on the basis employed in calculations.

Approximations to the exact (in the Hartree–Fock approximation) separable embedding potential were introduced in [490] that enable one to incorporate this potential into existing molecular calculation packages. The test calculations on the (CH3)2O molecule were performed that showed good accuracy of the potential.

So far, the formulation of pseudopotentials has been strictly nonrelativistic, so that the issue of heavy elements still remains to be addressed. In the next section we consider relativistic pseudopotentials.

310 8 Basis Sets and Pseudopotentials in Periodic LCAO Calculations

Table 8.4. The dipole moments (a.u) [488]

	Theor.1a	Theor.2b	Theor.3c	Expt.
SiH3F	1.226	0.555	0.527	0.511
SiH3Cl	0.899	0.644	0.631	0.515
SiH3Br	0.267	0.502	0.533	0.519
SiH3I	0.257	0.536	0.560

aThe bond is saturated with the hydrogen

bThe bond is saturated with the pseudosilicon cAll-electron Hartree–Fock calculations

Table 8.5. The charge on atom A, A= H, F, Cl, Br, I, [488]

	Theor.1a	Theor.2b	Theor.3c
SiH4	0.000	–0.157	–0.149
SiH3F	–0.380	–0.523	–0.687
SiH3Cl	–0.278	–0.398	–0.416
SiH3Br	–0.032	–0.035	–0.040
SiH3I	–0.097	–0.263	–0.352

aThe bond is saturated with the hydrogen

bThe bond is saturated with the pseudosilicon

cAll-electron Hartree–Fock calculations

8.3Relativistic E ective Core Potentials and Valence Basis Sets

8.3.1 Relativistic Electronic Structure Theory: Dirac–Hartree–Fock and Dirac–Kohn–Sham Methods for Molecules

Heavy-element systems are involved in many important chemical and physical phenomena. However, they still present di culties to theoretical study, especially in the case of solids containing atoms of heavy elements (with the nuclear charge Z ≥ 50). In this short description of relativistic electronic-structure theory for molecular systems we follow [496] and add a more detailed explanation of the Dirac–Kohn–Sham (DKS) method. For a long time the relativistic e ects underlying in heavy atoms had not been regarded as such an important e ect for chemical properties because the relativistic e ects appear primarily in the core atomic region. However, now the importance of the relativistic e ects, which play essential and vital roles in the total natures of electronic structures for heavy-element molecular and periodic systems, is recognized [496].

To treat the relativistic e ect theoretically, the Dirac Hamiltonian should be applied instead of the nonrelativistic Schr¨odinger Hamiltonian. The Dirac one-particle Hamiltonian has the form

ˆ	2	+ V (r)	(8.36)
hD = cα · pˆ + (β − 1)c