A MinMax self-consistent-field approach for auxiliary

THE JOURNAL OF CHEMICAL PHYSICS 130, 114106 共2009兲

A MinMax self-consistent-field approach for auxiliary density functional theory Andreas M. Köster,a兲 Jorge M. del Campo, Florian Janetzko, and Bernardo Zuniga-Gutierrez Departamento de Química, CINVESTAV, Avenida Instituto Politécnico Nacional 2508, A.P. 14-740, Mexico D.F. 07000, Mexico

共Received 8 October 2008; accepted 15 January 2009; published online 19 March 2009兲 A MinMax self-consistent-field 共SCF兲 approach is derived in the framework of auxiliary density functional theory. It is shown that the SCF convergence can be guided by the fitting coefficients that arise from the variational fitting of the Coulomb potential. An in-core direct inversion of the iterative subspace 共DIIS兲 algorithm is presented. Due to its reduced memory demand this new in-core DIIS method can be applied without overhead to very large systems with tens of thousands of basis and auxiliary functions. Due to the new DIIS error definition systems with fractional occupation numbers can be treated, too. © 2009 American Institute of Physics. 关DOI: 10.1063/1.3080618兴 I. INTRODUCTION

The separation of Coulomb and exchange-correlation energy contributions in Kohn–Sham1,2 density functional theory 共DFT兲 implementations has inspired a wide variety of approximations for these energy terms.3–14 Originally, Baerends et al.3 suggested a least squares fit of the orbital density in the framework of the X␣ method in order to improve computational performance. Sambe and Felton4 used a similar approach for the linear combination of Gaussian-type orbital 共LCGTO兲 implementation of the X␣ method. Later on Dunlap et al.5 developed a variational fitting procedure for the Coulomb potential, which requires only three-center integrals. Today, this approximation is employed in almost all Kohn–Sham implementations which are based on the LCGTO expansion. By the work of Vahtras et al.8 the variational fitting of the Coulomb potential became also an active research field in the wave-function community 共see, for example, Refs. 15 and 16 and references therein兲. Here it is named resolution of the identity. This approximation reduces the formal scaling of the Coulomb integral calculation from N4 to N2 ⫻ M, where N and M denote the number of basis and fitting functions, respectively. The number of fitting functions is typically three to five times the number of basis functions. It has been shown in the literature17 that the variational fitting of the Coulomb potential is robust and that its accuracy is within the intrinsic accuracy of LCGTO-DFT methods. It is important to note that the variational nature of the approximate energy expression is the key to success for this particular approximation. Because the energy stays variational in this approach analytic gradients and higher energy derivatives can be formulated.18–23 More recently, it has been shown that the energy expression remains variational even if the approximated density from the variational fitting of the Coulomb potential is used for the calculation of the exchange-correlation energy and potential.11 In this approach the Kohn–Sham potential is expressed over the auxiliary a兲

Electronic mail: [email protected].

0021-9606/2009/130共11兲/114106/8/$25.00

density and, therefore, it is named auxiliary density functional theory 共ADFT兲. With ADFT reliable first-principles calculations on systems with thousands of basis functions are possible.24,25 Recently, a modified form of McWeeny’s selfconsistent perturbation theory26–28 has been formulated within the ADFT framework,29 too. Different from the usual implementation of coupled-perturbed Kohn–Sham equations a noniterative solution was suggested. Due to the fitting of the Coulomb potential two sets of coefficients, the molecular orbital 共MO兲 and density fitting coefficients, occur in the self-consistent-field 共SCF兲 procedure of ADFT. The variational fitting is performed as described in Ref. 5. During a SCF step the density fitting coefficients are calculated by minimizing the error E2 defined as E2 =

1 2

冕冕

⌬␳共r1兲⌬␳共r2兲 dr1dr2 . 兩r1 − r2兩

共1兲

Here ⌬␳共r兲 represents the difference between the orbital density ␳共r兲 and the auxiliary density ˜␳共r兲 which, in our case, is expanded in Hermite Gaussian functions.30,31 So far the variational fitting of the Coulomb potential is mainly described as an integral approximation in the literature. Instead, in this paper we analyze the influence of this approximation on the SCF procedure. In particular, we show that the SCF convergence, usually altered by the manipulation of MO coefficients or density matrices, can also be altered via the fitting coefficients, i.e., the SCF convergence can be driven by them. Because the fitting coefficients are vector rather than matrix quantities efficient SCF convergence accelerations for very large systems can be developed. For this purpose we will rederive Dunlap’s fitting equation system directly from the ADFT energy expression. This derivation shows that the SCF optimization employing the variational fitting of the Coulomb potential is not a minimization but instead a MinMax optimization.32–35 The article is organized in the following manner. In Sec. II the fitting equation system for the Coulomb potential is

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rederived from the approximated energy expression. The MinMax character of the SCF optimization is analyzed on a model system with two orbitals and two auxiliary functions. In Sec. III convergence acceleration methods for the MinMax SCF are discussed. A new incore implementation for the direct inversion of the iterative subspace 共DIIS兲 method is derived. After a brief description of the computational methodology the performance of the described SCF convergence acceleration methods is discussed in Sec. IV Concluding remarks are drawn in Sec. VI.

II. THE MinMax SCF APPROACH

For the description of the variational fitting of the Coulomb potential and the corresponding SCF procedure the non-spin-polarized notation in combination with a local exchange-correlation functional is used for simplicity. The total electronic energy expression in a LCGTO-DFT approach is given by 1 E = 兺 P␮␯H␮␯ + 具␳ 储 ␳典 + Exc关␳兴. 2 ␮,␯

=

冕冕

␳共r1兲␳共r2兲 dr1dr2 兩r1 − r2兩

1 兺 兺 P␮␯P␴␶具␮␯ 储 ␴␶典, 2 ␮,␯ ␴,␶

共3兲

where ␮, ␯, ␴, and ␶ indicate atomic orbitals. The symbol 储 denotes the Coulomb operator 1 / 兩r1 − r2兩. In the framework of LCGTO-DFT methods the classical Coulomb energy can be expressed over four-center two-electron integrals 具␮␯ 储 ␴␶典, analog to Hartree–Fock methods. The last term Exc关␳兴 in Eq. 共2兲 represents the exchange-correlation energy which is usually calculated by numerical integration. An exception is the recently developed analytical DFT method14 in which this term is calculated analytically. In any case, the described MinMax SCF approach holds for all flavors of DFT methods that employ the variational fitting of the Coulomb potential independent of how the exchange-correlation energy is calculated. The orbital density appearing in Eq. 共3兲 is given in terms of the atomic orbitals and the density matrix elements

␳共r兲 = 兺 P␮␯␮共r兲␯共r兲, ␮,␯

˜␳共r兲 = 兺 x¯k¯k共r兲.

共6兲

¯k

Here ¯k共r兲 represents a Hermite Gaussian auxiliary function and x¯k the corresponding density fitting coefficient. According to the original work of Dunlap et al.5 the fitting coefficients are obtained during the SCF procedure by minimization of the fitting error E2, 1 E2 = 具␳ − ˜␳ 储 ␳ − ˜␳典 2 1 1 = 具␳ 储 ␳典 − 具␳ 储 ˜␳典 + 具˜␳ 储 ˜␳典 2 2 1 1 ¯ 储 ¯l典. = 具␳ 储 ␳典 − 兺 兺 P␮␯具␮␯ 储 ¯k典x¯k + 兺 x¯kx¯l具k 2 2 ␮,␯ ¯ ¯¯ k

k,l

共7兲

共2兲

Here ␮ and ␯ denote atomic orbitals, P␮␯ is the associated density matrix element, and H␮␯ is a core Hamiltonian matrix element including kinetic energy and nuclear attraction. The second term in Eq. 共2兲 represents the classical two-electron Coulomb energy 1 1 具␳ 储 ␳典 = 2 2

To circumvent the evaluation of four-center two-electron integrals in the above energy expression an auxiliary density ˜␳ is introduced,

共4兲

with

Due to the positive definite nature of E2, the following inequality holds: 1 1 具␳ 储 ␳典 ⱖ 具␳ 储 ˜␳典 − 具˜␳ 储 ˜␳典. 2 2

共8兲

Using this inequality and the approximated density for the calculation of the exchange-correlation contribution the ADFT energy expression is obtained,11 1 E = 兺 P␮␯H␮␯ + 具␳ 储 ˜␳典 − 具˜␳ 储 ˜␳典 + Exc关˜␳兴 2 ␮,␯ = 兺 P␮␯H␮␯ + 兺 兺 P␮␯具␮␯ 储 ¯k典x¯k ␮,␯

−

␮,␯ ¯k

1 兺 x¯kx¯l具k¯ 储 ¯l典 + Exc关˜␳兴. 2 ¯¯

共9兲

k,l

The ADFT energy 共9兲 approaches the exact energy 共2兲 with increasing size of the auxiliary function set. Because only relative energies are significant rather fast convergence is achieved. Different from the original energy expression 共2兲 the ADFT energy depends on both orbital and approximated densities. The differentiation of the ADFT energy with respect to the density matrix elements, keeping the fitting coefficients constant, yields the ADFT Kohn–Sham matrix elements, K ␮␯ =

冉 冊 ⳵E ⳵ P ␮␯

x

= H␮␯ + 兺 具␮␯ 储 ¯k典共x¯k + z¯k兲,

共10兲

¯k

with11 occ

P ␮␯ = 2 兺 c ␮ic ␯i . i

The c␮i and c␯i denote MO coefficients.

共5兲

¯ 储 ¯l典−1具l¯兩v 关˜␳兴典. z¯k = 兺 具k xc

共11兲

¯l

The solution of the corresponding Kohn–Sham equation

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共12兲

Kc = Sc␧

represents an energy minimization with respect to the MO coefficients under the MO orthogonality constraint. For our further discussion it is important to note that the density fitting coefficients are kept constant during this variation. As already mentioned the determination of the density fitting coefficients in the SCF procedure is based on the minimization of the fitting error E2

冉 冊 兺 冉 冊 ⳵ E2 ⳵ x¯k

=−

␮,␯

P

⳵ E2

P␮␯具␮␯ 储 ¯k典 + 兺 x¯l具l¯ 储 ¯k典 = 0 ∀ ¯k ,

1

0

x

(2) ELower

=E

`

c(2) , x(1)

´ ` (1) E = E c(1) , x(1) ´ U pper

` ´ (2) EU pper = E c(2) , x(2)

-1

共13兲

¯l

-2

2

= 具l¯ 储 ¯k典.

⳵ x¯l ⳵ x¯k

共14兲

` ´ (1) ELower = E c(1) , x(0)

˘ (0) (0) ¯ c ,x

P

-3

The MO coefficients and, thus, the density matrix are kept constant during this variation. Because the auxiliary function Coulomb matrix G with the elements G¯l¯k = 具l¯ 储 ¯k典 is positive definite the above variation is indeed a minimization of the fitting error E2. As a result, the following solution for the auxiliary function fitting coefficients is obtained: 共15兲

x = G−1J, with J¯k = 兺 P␮␯具␮␯ 储 ¯k典.

共16兲

␮,␯

Due to the variational nature of the approximated Coulomb energy in Eq. 共9兲 the density fitting coefficients can also be obtained directly from the variation in the ADFT energy keeping the density matrix constant,

冉 冊 ⳵E ⳵ x¯k

P

= 兺 P␮␯具␮␯ 储 ¯k典 − 兺 x¯l具l¯ 储 ¯k典 = 0 ∀ ¯k . ␮,␯

共17兲

¯l

The corresponding second energy derivative yields

冉

⳵ 2E ⳵ x¯l ⳵ x¯k

冊

= − 具l¯ 储 ¯k典.

共18兲

P

Because of the positive definiteness of the auxiliary function Coulomb matrix the energy variation with respect to the density fitting coefficients represents a maximization of the energy. Thus, the minimization of the fitting error 共13兲 corresponds to a maximization of the ADFT energy expression 共9兲. In Fig. 1 a contour map of the SCF energy surface of a model system with two orbitals and two fitting functions is depicted.35 Because of the orthonormality constraint for the MOs and the applied charge conservation constraint for the integrated approximated density only two degrees of freedom exist: one MO coefficient and one fitting coefficient. They are denoted by c and x in Fig. 1. The orthogonality of the corresponding axes in Fig. 1 represents the decoupling of the variable spaces in the SCF iterations. In this figure the individual steps during SCF iterations are shown, too. Starting from the black point in the lower left corner with the variables 兵c共0兲 , x共0兲其 an energy minimization 共lower horizontal red line兲 is performed by solving the corresponding Kohn–Sham equations. This represents a minimum search

-0.5

-0.25

0

0.25

0.5

c

0.75

1

c

FIG. 1. 共Color兲 A contour diagram of the SCF energy surface of a 2 ⫻ 2 model system. The red and blue dots represent lower and upper energy bounds of the corresponding SCF cycles, respectively. The red and blue lines represent energy minimization and maximization steps along isolines of constant fitting x and constant orbital c coefficients, respectively. The colors denote the SCF energy which increases from light green to red.

along an isoline for constant fitting coefficients. The result共1兲 ing energy 共lower right red point in Fig. 1兲 is named Elower = E共c共1兲 , x共0兲兲 in order to emphasize that it represents the lower energy bound for this SCF step. From the new MO coefficients a new density matrix is built and the corresponding fitting coefficients are obtained by minimizing the fitting error E2, i.e., by solving equation system 共13兲. As already discussed above, this represents a maximization of the corresponding energy expression. Because the MO coefficients are kept constant the energy maximization is performed along an isoline for constant orbital coefficients. This corre共1兲 . The sponds to the vertical blue line emerging from Elower result of this energy maximization is an upper energy bound 共1兲 Eupper = E共c共1兲 , x共1兲兲 for this SCF step 共upper right blue point in Fig. 1兲. With the calculation of Eupper the current SCF step finishes and the described procedure is repeated until SCF convergence is reached. In the neighborhood of the SCF convergence point holds E共cⴱ,x兲 ⱕ E共cⴱ,xⴱ兲 ⱕ E共c,xⴱ兲.

共19兲

Here the stars denote converged variable sets. These inequalities define a MinMax optimization problem.32–34 The convergence point can either be reached by minimization of the upper energies 共blue points in Fig. 1兲 of each SCF step or by maximization of the lower energies 共red points in Fig. 1兲 of each SCF step. Both paths join in the SCF convergence point. The iterative nature of the SCF procedure guarantees that despite the artificial separation of the variable spaces, a converged solution for both variable sets, MO and density fitting coefficients, is obtained with the SCF convergence. Due to the separation of the variable spaces the upper and lower energy bounds of each SCF step are no global bounds. Nevertheless, the energy difference between these bounds

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共I兲 共I兲 ⌬⑀ = Eupper − Elower

共20兲

can be used as a reliable SCF convergence measure. This difference is always positive 共see Appendix A for a detailed discussion兲 and its value is a direct measurement of how far the current SCF step is away from the convergence point. Therefore, it is desirable to calculate Elower and Eupper in each SCF step and use them to guide the SCF convergence.

III. CONVERGENCE ACCELERATION OF MinMax SCF

The density fitting coefficient mixing 共I兲

共I兲

x ← ␣x + 共1 − ␣兲x

共I−1兲

冉 冊冉 冊 冉 冊 B 1 1† 0

0 c = . 1 −␭

共22兲

The B matrix elements BIJ = 具e共I兲 兩 e共J兲典 collect the error vector products with a suitable defined metric in the error space. The Lagrange multiplier ␭ arises from the normalization constraint of the coefficients cI. Several other equivalent formulations exist.39 We will use here the original definition.37 However, generalization to other formulations is straightforward. A suitable error quantity for DIIS is the commutator of the density and Kohn–Sham matrix which must vanish at SCF convergence.28 In a nonorthogonal atomic orbital basis this commutator is given by e共I兲 = K共I兲P共I兲S − SP共I兲K共I兲 .

共23兲

共21兲

has no effect on the solution of the Kohn–Sham equations, i.e., on the minimization of the SCF energy. It only reduces the uphill step size of the energy maximization 共vertical blue line in Fig. 1兲. The mixing coefficient ␣ is defined in the open interval 共0,1兲. In practice a lower bound of 0.1 is used in deMon2k.36 The smaller is ␣ chosen, the smaller is the uphill step. This slows down the SCF convergence but avoids overstepping of the SCF convergence point that may occur for too large ␣ values. The lower energy bounds resulting from the solution of the Kohn–Sham equations that are indicated by the red dots in Fig. 1 are the reference energies for the described SCF with density fitting coefficient mixing. Therefore, the converged SCF energy is approached from below. On the other hand, if a density matrix mixing 共Hartree damping兲 is employed the upper energy bounds 共blue points in Fig. 1兲 become the reference energies. In this case the downhill energy minimization step 共red horizontal line in Fig. 1兲 is cut. The uphill step is then performed unmodified and the resulting upper energy bounds indicated by the blue dots in Fig. 1 approach from above the converged SCF energy. This discussion reveals that density fitting coefficient mixing and density matrix mixing have similar effects on the SCF convergence. They are, however, not equal. Moreover, a simultaneous mixing of both variable sets is not possible because the reference energy points are then lost. In deMon2k we decided to use the density fitting coefficient mixing instead of density matrix mixing because the density fitting coefficients represent a vector rather than a matrix quantity. A more sophisticated tool to improve the SCF convergence is the DIIS, as suggested by Pulay.37,38 Particularly, at the end of the SCF convergence DIIS has proven to be highly efficient showing nearly quadratic convergence. In the DIIS method an error vector e共I兲, where I is the step index, is constructed in each SCF step. The error vector must be related to the gradient of the electronic energy with respect to the SCF parameters and thus vanishes for the SCF solution. As parameters, the elements of the density matrix or the Kohn–Sham matrix are used. As discussed in Ref. 37, the least-squares criterion, together with the condition that the expansion coefficients cI must be normalized, leads to a small set of linear equations,

The BIJ matrix elements can then be calculated as BIJ = Tr共e共I兲e共J兲兲.

共24兲

With these matrix elements Eq. 共22兲 is solved and a new 共extrapolated兲 Kohn–Sham matrix is constructed from the DIIS solution DIIS 共I兲

K =

兺J cJK共J兲 .

共25兲

Several varieties of the above described DIIS implementation exist in the literature.40–45 They are used with great success in many electronic structure programs. However, in parallel applications46 or for very large systems with thousands of basis functions the above described DIIS algorithm is not without drawbacks. Whereas the parallel bottleneck can be overcome in conventional DIIS implementation the memory bottleneck is much harder to address. The storage of the density and Kohn–Sham matrices becomes for large systems an I / O bottleneck in the SCF iterations and, thus, jeopardizes the computational benefit of the DIIS procedure. In particular, for large scale ADFT calculations this represents a severe problem. Therefore, an alternative formulation for the DIIS algorithm is necessary. Based on our previous discussion of the density fitting coefficient mixing it is rather obvious that the guidance of the SCF convergence by the density matrix can be substituted by a guidance through density fitting coefficients. Following this idea we introduce a new error vector for the DIIS algorithm. From the MinMax condition 共19兲 follows that the vanishing of the density fitting coefficient’s gradient at the lower reference energies 共red dots in Fig. 1兲

冉 冊 共I兲 ⳵ Elower ⳵ x¯k

P

共I兲 = 兺 P␮共I兲␯具␮␯ 储 ¯k典 − 兺 x¯l 具l¯ 储 ¯k典,

␮,␯

共26兲

¯l

is a necessary and sufficient condition for the MinMax SCF convergence. After solving for the MOs at each DIIS step, the density matrix is eliminated from this expression by employing the fitting equation system 共13兲,

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TABLE I. The SCF convergence of TiCp2Cl2 in its experimental geometry as depicted in Fig. 2. For each SCF cycle the mixing coefficient, DIIS dimension, and determinant as well as the corresponding MinMax error are given.

SCF cycle

Mixing

DIIS dimension

DIIS determinant

MinMax error 共a.u.兲

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18

0.30 0.30 0.30 0.30 0.30 0.30 0.20 0.13 0.13 0.13 0.13 ¯ ¯ ¯ ¯ ¯ ¯ ¯

¯ ¯ ¯ ¯ ¯ ¯ ¯ ¯ ¯ ¯ ¯ 2 3 4 5 5 4 3

¯ ¯ ¯ ¯ ¯ ¯ ¯ ¯ ¯ ¯ ¯ 1.369⫻ 10−4 2.404⫻ 10−7 2.082⫻ 10−10 1.987⫻ 10−14 4.797⫻ 10−14 3.439⫻ 10−13 3.061⫻ 10−11

2282.3369 14.1279 4.8753 1.8911 0.1449 0.1367 0.2180 0.2731 0.0223 0.0014 9.6⫻ 10−4 6.8⫻ 10−4 7.2⫻ 10−5 6.3⫻ 10−5 3.3⫻ 10−6 4.4⫻ 10−7 2.1⫻ 10−7 1.5⫻ 10−8

共I+1兲 P␮共I兲␯具␮␯ 储 ¯k典 = 兺 x¯l 具l¯ 储 ¯k典. 兺 ␮,␯ ¯

共27兲

l

Substituting Eq. 共27兲 into Eq. 共26兲 yields the following new DIIS error vector element definition: e共I兲 k

冉 冊

共I兲 ⳵ Elower = ⳵ x¯k

P

=兺 ¯l

共I+1兲 共x¯l

−

共I兲 x¯l 兲具l¯ 储 ¯k典.

共28兲

The B matrix elements are then calculated as the scalar products of these error vectors, BIJ = e共I兲 · e共J兲 .

共29兲

At this point it should be noted that the ADFT Kohn–Sham matrix 共10兲 depends only on the fitting coefficients and not on the density matrix. Therefore, the extrapolated fitting coefficients from the solution of the DIIS 关Eq. 共22兲兴 DIIS

x共I兲 =

兺J cJx共J兲

共30兲

determine completely the ADFT Kohn–Sham matrix. As a result a matrix free incore DIIS algorithm applicable to very large systems is realized. IV. COMPUTATIONAL METHODOLOGY

All calculations were performed with the LCGTO-DFT code deMon2k.36 The Coulomb energy was calculated by the variational fitting procedure proposed by Dunlap et al.5 The resulting auxiliary density was then used for the calculation of the exchange-correlation energy and potential, too. Thus, all calculations were performed in the framework of ADFT employing a double zeta valence plus polarization basis set in combination with the A2 auxiliary function set.47 In all

FIG. 2. 共Color online兲 Geometrical structures of the test systems for the MinMax SCF convergence. From top left to bottom right: TiCp2Cl2, V共CO兲6, C240, Si118H84O198Al2Na, C320, and Si288O600H176.

calculations the Dirac exchange48 in combination with the Vosko et al.49 correlation functional was employed. The exchange-correlation energy and potential were numerically integrated on an adaptive grid50,51 with a grid accuracy of 10−5. If not otherwise stated a tight-binding start density in combination with the deMon2k default SCF convergence criterion of 10−5 a.u. for the MinMax energy error was employed. The DIIS procedure was activated after the MinMax energy error dropped below 10−3 a . u.; again this is a deMon2k default setting. For structure optimizations a quasiNewton restricted step method52 in delocalized internal coordinates53,54 with an automatized selection of relevant primitive coordinates55 was used. The convergence of the structure optimization was based on the analytic energy gradient and displacement vectors with thresholds of 10−4 and 10−3 a . u., respectively. V. APPLICATIONS

The SCF convergence of an ADFT calculation of TiCp2Cl2, with Cp denoting cyclopentadienyl 共C5H5兲 substituents, is listed in Table I. For this calculation the experimental structure56 for TiCp2Cl2 was employed 共Fig. 2兲 and the MinMax energy error convergence criterion was tightened to 10−7 a . u. in order to discuss the performance of the new incore DIIS in more details. Table I lists the mixing coefficients, DIIS dimensions, and determinants as well as the MinMax energy error of each SCF cycle. As this table shows the SCF iteration starts with simple relaxation em-

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TABLE II. Total number of SCF cycles and DIIS cycles for selected systems. The number of basis and auxiliary functions is denoted by N and M, respectively. The structures are depicted in Fig. 2. System V共CO兲6 C240 C320 Si118H84O198Al2Na Si288O600H176

N

M

SCF cycles

DIIS cycles

206 3600 4800 5689 15 352

463 8160 10 880 10 731 34 064

19 14 14 15 17

4 4 4 5 3

selected them in order to demonstrate that the performance of the incore DIIS is almost system size independent. For this reason we also list the number of basis and auxiliary functions N and M, respectively, in Table II, too. The largest system possesses more than 15 000 basis functions and converges within 17 SCF cycles. Even for this system the memory demand of the described incore DIIS is negligible compared to the size of the SCF matrices. Thus, all incore DIIS relevant data can be kept in random access memory. VI. CONCLUSIONS

ploying the deMon2k default mixing coefficient of 0.3. In the first six steps the MinMax error decreases gradually. In steps seven and eight an increase in the MinMax error can be observed. As counteraction the mixing coefficient is automatically decreased to 0.2 and 0.13, respectively. Further relaxation then reduces the error to 9.6⫻ 10−4 in SCF step 11. From here on the incore DIIS algorithm is enabled. In the following SCF steps the DIIS space is enlarged to up to five error vectors in SCF step 15. At the same time the DIIS determinant that is the determinant of the coefficient matrix in Eq. 共22兲 reduces to around 10−14, indicating more and more linear dependencies in the DIIS space. In these four SCF steps the MinMax error reduces by more than two orders of magnitude to 3.3⫻ 10−6 a . u. and, therefore, superlinear SCF convergence is achieved. A similar reduction in the SCF error would take more than ten SCF steps with simple relaxation. When a representative DIIS error space is built, which happened in our example around SCF step 14, the incore DIIS shows near quadratic convergence, i.e., the MinMax energy error decreases by one order of magnitude in each SCF step. However, the reduction in the error space due to linear dependencies deteriorates this quadratic convergence at the end of the SCF iterations, as Table I shows. Nevertheless, the SCF convergence acceleration by the incore DIIS algorithm is remarkable. It usually reduces the necessary SCF cycles by more than a factor of two. The number of necessary SCF and DIIS cycles for some selected system single point calculations is listed in Table II. Except for Si288O600H176 all structures are optimized. The structures of these systems are depicted in Fig. 2. For V共CO兲6 an octahedral structure is optimized in accordance with the experimentally observed dynamical Jahn–Teller effect57 that was more recently confirmed by temperature dependent infrared and Raman spectroscopy.58 In order to simulate this effect in the structure optimization an Oh starting structure was employed and fractional occupation of a threefold degenerated orbital was enforced. Different from standard DIIS algorithms the new incore DIIS algorithm can be applied to fractional occupation, too, because the incore DIIS error is based on the density fitting coefficient error and not on the commutator between the density and Kohn–Sham matrix. As Table II shows the fractional occupied V共CO兲6 SCF calculation converges in 19 iterations with the last four being DIIS interpolations. The SCF convergence of V共CO兲6 is similar to other integer occupied transition metal carbonyls. Thus, fractional occupation does not alter significantly the performance of the incore DIIS algorithm. The other systems in Table II are considerably larger than V共CO兲6. We

The equation system for the variational fitting of the Coulomb potential is rederived from the ADFT energy expression. From this derivation it is shown that the minimization of the fitting error corresponds to a maximization of the ADFT energy. As a consequence a MinMax SCF procedure is suggested. Various methods for the stabilization and acceleration of the SCF convergence are discussed in the context of this new MinMax SCF. In general, it is shown that SCF convergence can be guided by the fitting coefficient in a similar way as by the MO coefficients or density matrix elements in traditional approaches. Because the density fitting coefficients form a vector rather than a matrix, memory efficient SCF convergence acceleration methods become feasible. A particular implementation of the well-known DIIS method on the basis of the density fitting coefficients is described and validated. This new incore DIIS algorithm shows super linear convergence and can be applied to fractional occupation, too. Because the I / O bottleneck of traditional DIIS algorithms is avoided the incore DIIS algorithm can be applied to systems with tens of thousands of basis and auxiliary functions. Test calculations with deMon2k are presented. ACKNOWLEDGMENTS

This work was financially supported by the CONACyT Project No. 60117-E. J.M.C. and B.Z.G. gratefully acknowledge CONACyT Ph.D. fellowships 共Grant Nos. 180545 and 200103兲. A.M.K. thanks P. Calaminici, M. E. Casida, A. Goursot, L. G. M. Petterson, J. U. Reveles, D. R. Salahub, and A. Vela for many valuable discussions during the MinMax SCF development. Special thanks to F. Schrade for his help in the elaboration of the original idea. APPENDIX A: ROBUST COULOMB FITTING

In order to ensure an always positive MinMax energy error a numerical stable solution for the fitting equation system 共15兲 is required. Particularly for large auxiliary function sets with high angular momentum indices the Coulomb matrix becomes bad conditioned and equation system 共15兲 becomes numerically instable. In such cases singular value decomposition 共SVD兲 is the method of choice for the solution of Eq. 共15兲. Because our numerically most stable implementation of SVD for the fitting equation system slightly deviates from the literature59 we describe it here in more details. The first step represents the diagonalization of the Coulomb matrix G,

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J. Chem. Phys. 130, 114106 共2009兲

MinMax SCF for ADFT

共A1兲

D = U†GU.

Here U denotes the orthogonal transformation matrix that contains the eigenvectors of G. For further discussion it is important to note that the diagonalization of G is equivalent to the transformation of the auxiliary functions into a diagonal basis with respect to the two-electron Coulomb operator. After the diagonalization of G all eigenvalues in D that are below a certain threshold are quenched, i.e., set to zero. The eigenvectors that correspond to these quenched eigenvalues are explicitly eliminated from the U matrix. As a result, U becomes rectangular and the dimensionality of the auxiliary function basis that is equivalent to the number of column vectors in U is reduced to a smaller dimension d, Dd = U†dGUd

共A2兲

˜ = U D U† . G d d d

共A3兲

and

The subscript d indicates matrices 共and vectors兲 with reduced ˜ denotes the SVD approximate to G. Indimensionality. G stead of constructing an approximate inverse to G we are transforming the fitting equation system 共15兲 into the reduced diagonal auxiliary function basis, ˜ x = J, G

共A4兲

UdDdU†dx = J,

共A5兲

DdU†dx = U†dJ,

共A6兲

with U†dx = xd ,

共A7兲

U†dJ = Jd ,

共A8兲

then follows 共A9兲

D dx d = J d .

The equation system is then solved in the diagonal representation 共A10兲

xd = D−1 d Jd . D−1 d

are diagonal the numerically delicate Because Dd and couplings in the solution of the equation system are removed. The resulting fit coefficients xd are finally back transformed to the original auxiliary function set representation by x = U dx d .

共A11兲

Even so the above described algorithm is formally identical to the direct calculation of the approximated inverse of G by SVD, it is numerically far more stable. This method is recommended if large auxiliary function sets and tight SCF convergences are requested. Typical examples are polarizability and frequency calculations. 1 2

W. Kohn and L. J. Sham, Phys. Rev. 140, A1133 共1965兲. P. Hohenberg and W. Kohn, Phys. Rev. 136, 864 共1964兲.

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