LIMITACIONES DEL ESTUDIO

LEDO-DFT

Due to the reasons outlined in Sec. 2.5, for large molecular systems the numerical errors introduced by the LEDO approximation can lead to an uncontrollable be- havior of the SCF process. It has already been stated that these errors can always be reduced by an improvement of the LEDO expansion basis or by using robust expressions for the approximated ERIs. In the following, however, a projection technique for the elimination of near-linear dependent AOs shall be introduced, which can be applied in order to guarantee for SCF convergence of LEDO-DFT calculations without resorting to the aforementioned remedies. Although AO ba- sis sets commonly used in MO calculations are linearly independent in the strict sense of the term it may happen that they are approximately linearly dependent from a numerical point of view [151]. As a measure for the degree of near-linear dependence it is convenient to choose the condition number of the overlap matrix S which represents the metric of the AO basis set {φµ} [211]. The condition num-

ber of the overlap matrix given by the ratio of the highest to the lowest eigenvalue is not invariant with respect to local orthogonalization, i.e., orthogonalization of the AOs located on a particular atom. However, the changes of the condition number due to local orthogonalization are comparatively small and do not affect the present discussion.

L¨owdin [151] approached the problem of near-linear dependence by consider- ing the canonically orthogonalized orbitals (COOs). Arranging the AO basis func- tions φµand the COOs χkas row vectors Φ = {φ1, φ2, . . . } and χ = {χ1, χ2, . . . },

respectively, the latter can be written as

χ = ΦUe−1/2. (2.142) U is a unitary matrix diagonalizing the overlap matrix S with the eigenvalues being arranged in a diagonal matrix e,

U†SU = e. (2.143)

The squares of the coefficients of the COOs do not add up to one as it would be the case for an orthogonal basis, instead one has

X µ      Uµk e1/2_k      2 = 1 ek . (2.144)

Near-linear dependence occurs if there are eigenvalues ek close to zero. According

to Eq. (2.144), the AO coefficients of the corresponding eigenvectors can become very large. These are just the eigenvectors which need to be eliminated from the basis if numerical problems are to be avoided.

The projection scheme will be outlined in the following. A threshold cmax for

the condition number is defined and, starting with the eigenvector belonging to the lowest eigenvalue, as many eigenvectors are removed as is necessary to fulfill the condition

eN

eM +1

< cmax (2.145)

if N is the dimension of the AO basis set and the eigenvalues ek are arranged in

increasing order. Thus eliminating M eigenvectors, the remaining ones are used to form a projection operator

Q =

N

X

k=M +1

|χkihχk| (2.146)

which acts on the KS operator HKS from the left and from the right, HKS,proj = Q†HKSQ =

N

X

k,l=M +1

|χkihχk|HKS|χlihχl|. (2.147)

Performing the projection in the AO basis, the projected KS matrix reads as HKS,proj = Q†HKSQ = SPHKSPS (2.148)

with the matrix elements of P being given as Pµν = N X k=M +1 UµkUνk∗ ek . (2.149)

Suhai et al. [212] made plausible why this projection scheme should dampen the effect of numerical inaccuracies in the evaluation of the KS matrix. First, it should be noted that the projection reduces the dimension of the basis from N to N − M as can be seen from Eq. (2.147) thus raising the electronic energy. However, this effect will be small because the eliminated basis vectors contribute mainly to the space of virtual orbitals. Following Suhai et al. [212], the KS matrix in the AO basis can be decomposed in an exact part and a contribution arising from numerical errors,

HKS= HKS,exact+ ∆HKS. (2.150) To clarify the effects of numerical errors it is necessary to transform the KS matrix from the non-orthogonal AO basis to an orthogonal basis. Choosing the basis of COOs one obtains

HKS,COO = e−1/2U†HKSUe−1/2. (2.151) Thus, the error part of the KS matrix is given by

∆H_klKS,COO = 1 e1/2_k e1/2_l

X

µ,ν

U_µk∗ Uνl∆HµνKS. (2.152)

It is easily recognized from Eq. (2.152) that the presence of small eigenvalues of the overlap matrix leads to an amplification of the inaccuracies in the KS matrix. Cycle after cycle the errors increase and the SCF procedure can become unstable. Thus, it is desirable to eliminate the eigenvectors belonging to such small eigenvalues.

The use of this technique is inexpensive and requires only minor modifications to the program.

Implementation

For the practical application the LEDO-DFT formalism including analytical gra- dients and the projection technique, which have been presented in the last chapter in Secs. 2.4 to 2.7, have been turned into a working implementation. In this chap- ter a brief overview of this implementation is given and some important details are highlighted. Possible and desirable extensions will be discussed in chapter 6.

3.1

Framework and General Features

All implementations presented in this work1 _{[188, 189] have been carried out in}

the framework of the program package TURBOMOLE [10, 11] in version 5.1. The programs RIDFT and RDGRAD have been chosen for the implementation of the energy and the gradient calculations, respectively. These are the programs for KS-DFT calculations within the RI approximation (conventional density fit- ting) and therefore appear as a natural choice for the implementation of LEDO- DFT, although the programs DSCF and GRAD could have been used as well. These latter programs allow to perform HF and unapproximated KS-DFT calcu-

1 _{The SCF part is based on a preliminary implementation of LEDO-DFT which had been}

realized in the framework of a Diplomarbeit (Master’s Thesis) [12]. It was restricted to Cartesian Gaussian functions as AO basis and the LEDO approximation was limited to the Hartree term. In this version the perturbative approach of Ahlrichs and the a priori elimination for the determination of the LEDO expansion coefficients (cf. Sec. 2.4.3), auxiliary orbitals to augment the LEDO expansion basis and integral prescreening have not been available.

lations. The projection technique has been implemented both into the program RIDFT and DSCF. The programming language FORTRAN 77 [213] has been used throughout, apart from the dynamical memory management which is han- dled by FORTRAN 90 [214] statements. All required quantities are completely kept in the core memory during the calculations. While this might not be the best choice for future applications, it is completely sufficient for the purposes of this work. Communication between the programs RIDFT and RDGRAD, which is necessary for efficient LEDO-DFT calculations, has been realized via files. Ex- tensive use of BLAS [215] and LAPACK [216] linear algebra routines is made whenever possible and the translational invariance of the ERIs [217–219] is fully exploited in the gradient program. Using a prescreening of the ERIs based on the Schwarz inequality [55, 220], it is possible to determine the LEDO expan- sion coefficients and derivatives thereof only for non-negligible overlap densities. Point group symmetry is not exploited in the present implementation. However, this is not considered as a major drawback since most large molecular systems of interest actually do not possess any symmetry elements other than C1.

The LEDO approximation has been implemented both for the Hartree and the XC contributions for energy and gradient calculations, with the option to restrict the LEDO approximation to the Hartree term. All other contributions to the KS matrix, the energy and the gradient, are treated conventionally as implemented in TURBOMOLE. The LEDO expansion basis {Ωp} can be augmented by a set

of auxiliary orbitals {ηµ} and the full Cartesian components (i.e. 6 d-type, 10 f -

type etc.) of the Gaussians are always used in the LEDO expansion. Apart from the modified Cholesky factorization [1], the perturbative approximation [127] or the a priori elimination [188] can be employed in combination with a standard Cholesky factorization for the determination of the LEDO expansion coefficients and their derivatives. Finally, the menu-driven input program DEFINE has been adapted in such a way that input files appropriate for LEDO-DFT calculations with TURBOMOLE can be created with ease.