Quantum Chemistry of Solids / 15-Electron Correlations in Molecules and Crystals

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which are in a way close to the centroid of the WF Wi(r). The criterion (5.107)) is utilized to fit the WF spanned by the basis functions within Si(f it) (the WF-AO fit domain) in a least-squares sense with respect to the untruncated function. Hence, minimizing the functional (5.107) the fitted WFs are

!	Cµi(f it)φµ(r)	(5.109)
Wi(f it)(r) =

µ Si(f it)

with the fitted coe cients C(f it)	, obtained by solving the linear equations
µi
µ i
Sµ(AOµ		)Cµ(fiit) =	Sµ(AOµ	)Cµ i	(5.110)
S(f it)			µ

The fitting according to (5.110) is formally equivalent to other fitting techniques like density fitting [146, 147], with the AOs from the WF-AO fit domain in the role of the auxiliary basis functions.

By successively adding coordination spheres of atoms and thus their atomic functions to the set Si(f it) and refitting the coe cients Cµ(fiit) one can finally obtain the fitted WF Wi(f it)(r) that to a given tolerance, approximates the exact WF Wi(r).

a chosen tolerance the number of atoms contributing significantly to the fitted
Within (f it)!
The !	!
WFs Wi	(r) is essentially smaller than that of the functions Wi(r).

The fitting procedure can be applied also to the NOLOs, once they are constructed. localization-fitting process for the NOLOs is most e cient when the set of atomic orbitals and WF-AO fit domain Si(f it) at both stages are chosen to be the same. Indeed, small values of the fitting functional Ii(f it) can be expected, if all the sites with a relatively large value of the NOLO are included in the corresponding WF-AO fit domain. Thus, it does not make sense to use domains Si(f it) smaller than Si(loc), since the latter guarantees small values for localized functions only outside its region. On the other hand, if the value for the fitting functional within the chosen WF-AO fit domain is not su ciently small, it is more e cient, when one does not just enlarge it (as in the case of WFs), but recalculates beforehand the reference NOLOs with the new set Si(loc) enlarged accordingly. Actually, only the LCAO coe cients of the NOLO (or WFs) Cµi can be refitted by minimizing the functional (5.107). The L¨owdin coe cients cannot be modified by the fitting (5.110), since the corresponding AO overlap matrix involved is the identity matrix. This means that the L¨owdin coe cients within the WF-AO fit domain are optimal with respect to the functional (5.107) providing its minimum. For orthogonal atomic-like basis sets (such as L¨owdin-orthogonalized AOs), the fitting process just corresponds to the truncation of the WF coe cients according to the chosen WF-AO fit domain. Constructing the NOLO in the case of orthogonal basis sets might be more e cient, since the localization functional (5.103) addresses the coe cients relative to the orthogonalized basis-set functions directly. The e ciency of the fitting procedure for the localized functions depends on redundancies carried in the corresponding basis sets, which are large in the case of highly overlapping basis sets. Summarizing the above, the method for obtaining ultralocalized nonorthogonal fitted functions implies the following. Once the symmetry of the localized orbitals has been determined, for every symmetry-unique function the variational procedure

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in the symmetrized basis according to the functional (5.103) is performed, followed by generating the fitted functions with limited support from the former by solving (5.110). The sets of the atomic orbitals Si(loc) and Si(f it) in these two steps are to be taken as the same. If the value of the fitting functional (5.107) does not drop below a predefined threshold, a new star of atoms is added and the process is repeated.

The atomic functions projected onto the orthogonal complement of the occupied space (PAOs), which represent in the framework of local correlation methods the virtual states (see Sect. 5.4.1) are also localized functions. Therefore, similar techniques could also be used to restrict the spatial extent of these PAOs. However, since PAOs are to be constructed by projecting out the space spanned by the fitted WFs or NOLOs (rather than the untruncated parental WFs or NOLOs), one can anticipate that PAOs will automatically possess a limited support, determined basically by the WF AO fit domains.

The e ciency of the method suggested was demonstrated in diamond crystal local MP2 calculations [157]. Since in the present version of the CRYSCOR code [117] a possible nonorthogonality of the localized functions, representing the occupied manifold is not taken into account, the local MP2 energy in diamond has been calculated using only orthogonal WFs with and without fitting. The weakand distant-pair distances were set to 2 ˚A. The threshold for PAOs coe cients was set to 10−2. Only the coe cients for the AOs from the first three stars of atoms were included in the transformation of the 4-index integrals (WF fitting was also performed within the first three stars of atoms). Local MP2 correlation energy and timings were then compared with the corresponding values of a reference calculation, which were obtained by employing quite a low screening threshold of 5 × 10−4 for the WF coe cients (smaller thresholds are hardly manageable, presently [109]). Evidently, the fitting of WF coe cients can improve the accuracy and/or reduce the computational cost of the local MP2 calculations for solids. The results of these calculations show: if fitted WFs are used, the estimate of the error in the calculated MP2 energy (the di erence between the calculated and reference energies) becomes substantially smaller, while the CPU time of the calculation remains about the same.

5.4.3 Symmetry Exploitation in Local MP2 Method for Periodic Systems

In local MP2 calculations of solids it is important to exploit the point symmetry, in order to confine calculations to an irreducible subset; savings of computer time and storage can be substantial in a number of cases of importance. As was discussed in the preceding subsection, the WFs symmetry can be taken into account on the step of WFs generation using the theory of induced representations of space groups and symmetry information about the Bloch states involved in the WFs generation. However, there is another possibility – a symmetrization of a set of constructed LWF before starting the local MP2 calculations. A set of LWF is a posteriori symmetrized and symmetry-adapted localized WFs (SALWF) are constructed. A rapid account of this technique was given in [109], while a more detailed analysis can be found in [201].

SALWFs are partitioned into subsets (f ) belonging to a class b, each transforming into itself under the operators of a local symmetry point-subgroup Ff of the full point group F, while the operations of the associated leftor right-cosets transform subset f into another equivalent one in the same class. As an example in the diamond crystal

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case (F = Oh) subset f consists of one LWF centered in the middle of the atom–atom bond (Ff = D3d). The LWFs class consists of four LWFs describing the upper valence band of four subbands. Formally, the symmetrization and the localization procedures are in principle independent: for a chosen localization criteria the LWFs symmetrized before construction have to be close to those symmetrized a posteriori their construction. In fact, the real localization of electron density may be su ciently complicated. In a-posteriori symmetrization the choice of Ff is generally not unique (in the simple case of diamond it was unique) and two possible cases may be mentioned: (i) The subsets are chosen so as to minimize the order of Ff ; (ii) Ff is chosen so that the representation of its elements in the SALWF basis set is maximally factorized according to the irreducible representations of Ff . Functions optimally localized under a given criterion may feature both properties (as is the case when a regular representation is possible), any one of them, or none of them [201]. For example, let us consider four nonsymmetrized LWFs defined by Bloch states of four upper valence bands in MgO crystal (the valence electron pairs related to the anions are supposed to be localized according to the Boys criterion [38]). In that situation, the localization criterion is just compatible with an sp3 hybridization but, as the local point-symmetry group of the atom is cubic and not tetrahedral, the set of orbitals is constrained to form a reducible representation of Oh with no privileged orientation. The LWFs fall into a situation not described by the previously mentioned cases. A similar behavior is displayed by “banana” orbitals in triple bonds, or core electrons in a variety of highly symmetric crystals.

For the scope of numerical applications, even in those cases where localization gives rise to definite symmetry properties (as in the diamond case), accurate symmetry equivalences are only reached under strong numerical conditions in the localization algorithm. This can be, in several cases, very computationally demanding and a more e cient shortcut to obtain accurate symmetry equivalence together with a good localization character is therefore desirable.

Due to the previously mentioned di culties of an a priori prediction of what kind of symmetry behavior will be compatible with good localization properties under e cient criteria, in [201] an a-posteriori strategy was adopted based on a “chemical” analysis of the problem, which is made easier by the fact that WFs are expressed as a linear combination of atomic orbitals (AOs); it represents a compromise between the two criteria previously mentioned. Starting from a set of localized WFs, the main steps of the presented scheme are:

(1)recognize if classes of symmetry-related single-WF subsets already exist;

(2)classify the remaining WFs, if possible, into atomic or bond WFs, according to the Mulliken population analysis. The extension to periodic systems of Mulliken population analysis introduced for molecules is considered in Chap. 9. The electron

density of LWF Wi0 is partitioned into atomic populations QiAM , assigned to di erent atoms in the primitive cells M . One can preclassify Wi0 as

a) Atomic LWF: an atom AM exists for which QAM > 0.9. This category includes “core” LWFs, but also LWFs associated with valence AOs in anions of very ionic systems. “Lone-pair” or “hydrogen-bond” LWFs may also enter this category.

b) Bond LWF: two near-by atoms AM, BN exist such that QiAM +QiBN ≥ 0.9, and

the ratio QiAM /QiBN is comprised between 1 (pure covalency) and 1/4 (high ionicity). c) Atypical LWF: Wi0 does not enter the previous categories.

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(3)using the previous information recognize the subsets, the site-symmetry subgroups Ff and the irreducible representations to which they belong;

(4)generate, and orthonormalize the new SALWFs.

A criterion based on a Mulliken analysis is adopted again to subdivide atomic LWFs associated to the same irreducible atom A into shell-subsets (without loss of generality, A can be assumed to be in reference 0 cell). The bond LWFs are the representatives of double and triple bonds, and are assigned to the (AO−BM )-subset. Thus, part of the LWFs, classified as SALWFs, only require numerical refinement, or the subset including all the atypical ones. All others are grouped in subsets S, either of a “shell” or of a “bond” type, each one comprising ns members.

Bond and atomic LWFs are characterized by one and two atomic centers, respectively. Using this information the subgroup Gf that leaves invariant the corresponding symmetry element (either a center or an axis in real space) is found. The irreducible representations of Gf are found and for each of them the representative matrices are first constructed and, then, used to build up the idempotent projector operators for the first row and the shift operators for all other rows. The details of this procedure can be found in [201, 202].

Finally, when all atomic and bond subsets have been symmetry adapted, they are orthonormalized. Orthonormalization is performed in two steps. Using a given net in reciprocal space, the nonorthogonal functions generated in the previously described steps are transformed into Bloch functions, orthonormalized using a L¨owdin scheme in each k-point of the net, and backtransformed into Wannier functions. This is accomplished using a straightforward modification of the Wannierization scheme reported in [203]. At this step,the resulting SALWFs are orthonormal just under Born–Von Karman cyclic conditions. If necessary, the functions can be corrected to satisfy accurate orthonormalization by integration over the whole real space. We discussed a method [201] for generating from a set of (generally nonsymmetric) localized Wannier functions (LWF), an equivalent set of orthonormal functions that are adapted not only to the translational, but also to the point symmetry of the crystalline structure, while maintaining good locality features. They are called therefore SALWFs (symmetry adapted localized Wannier functions). At variance with other methods based on the induced representations theory this method is largely based on a chemical analysis of the structure of LWFs: despite this somewhat empirical character, it performs very well with a variety of molecular and crystalline systems.

The e ectiveness of this procedure was demonstrated [201] by way of examples, with reference to four test cases: the CH3Cl molecule (with C3v symmetry), linear polyacetylene, lithium bromide with the rocksalt structure (symmorpic space group Oh5 ) and crystalline silicon (nonsymmorphic space group Oh7 ). SALWFs generated appear well suited for use in the local correlation methods. In future, the symmetrized fitted LWFs can be used to have more a e cient computational procedure of LMP2 calculations.

Concluding the consideration of the modern electron-correlation methods for periodic systems we see that the first-principles LCAO post-HF methods for solids are essentially more di cult compared with those for the molecular systems. This is why until now the simpler semiempirical approaches have been popular for crystals. These approaches allow the intra-atomic and partly interatomic correlation e ects to be taken into account by introduction of semiempirical parameters calibrated on the ba-

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sis of first-principles calculations or experimental data. In many cases these methods give correct qualitative characteristics of the crystalline electronic structure that may be later corrected in the first-principles calculations.

The semiempirical LCAO methods for periodic systems are considered in the next chapter.