Applied density functional theory and the deMon codes 1964–2004

q 2005 Elsevier B.V. All rights reserved. Theory and Applications of Computational Chemistry: The First Forty Years Edited by C. Dykstra et al.

1079

CHAPTER 38

Applied density functional theory and the deMon codes 1964 – 2004 D.R. Salahub1, A. Goursot2, J. Weber3, A.M. Ko¨ster4 and A. Vela4 1

University of Calgary, 2500 University Drive NW, Calgary, Alberta, Canada, T2N 1N4 UMR 5618 CNRS, Ecole Nationale Supe´rieure de Chimie, 8 rue de l’Ecole Normale, 34296 Montpellier, Ce´dex 5, France 3 De´partement de Chimie Physique, Universite´ de Gene`ve, Sciences II, 30 quai Ernest Ansermet, 1211 Gene`ve 4, Switzerland 4 Departamento de Quı´mica, Cinvestav, Avenida Instituto Polite´cnico Nacional 2508, A.P. 14-740 Me´xico DF 07000, Me´xico

2

Abstract Advances in density functional theory and its applications over the past four decades are reviewed from the perspective of developers of the methodology and codes embodied in the deMon software. 38.1 INTRODUCTION. FROM THE 1920s TO THE 1960s One can consider that applied DFT goes back to the early 1930s when Dirac [1] and Wigner and Seitz [2,3] treated the problem of the exchange potential in the Thomas – Fermi atom [4 –7], arriving at the expression for local exchange, proportional to the 1/3 power of the density. A few years later, Slater [8] proposed a model of spherically symmetric atomic potentials embedded in a region of constant potential expanded in plane waves, which was later called the augmented plane wave (APW) method for the calculation of energy bands in solids. After the long gap generated by World War II, a new route to applications was opened by Slater’s 1951 paper [9], which introduced the idea of approximating the complicated non-local Hartree – Fock exchange operator by an average local potential. Slater’s derivation, which represents a generalization and extension of Wigner and Seitz, defined the properties of the exchange charge density associated with an electron and used the free-electron-gas model to approximate the exchange potential in terms of the local electron density. References pp. 1092– 1097

1080

Chapter 38

The magnitude of Slater’s exchange term in the Hamiltonian was questioned three years later by Gaspar [10] who, through a different derivation, obtained the same r1=3 form, but with a factor of 2/3. This new value was confirmed later by Kohn and Sham. The factor under debate, then named a, was used later on as an atom-dependent parameter [11] in the so-called Xa methodologies. The first among these was derived from the original muffin-tin approximation of Slater [8] and generalized by Johnson [12, 13] to treat molecular clusters. The multiple scattering-Xa (MS-Xa) method, also known as the Xa-Scattered Wave (Xa-SW) method is thus based on the local Hartree – Fock – Slater (HFS) approximation, with eigensolutions obtained using the multiple-scattering approximation. This was an important step as it represented one of the first attempts to adapt a method of theoretical solid-state physics to the study of molecular systems. From a more technical point of view the casting of the Hartree –Fock equations into eigenvalue equations, the so-called Roothan –Hall equations [14,15], was of fundamental importance for the future development of quantum chemical codes. The well-defined structure of the mathematical problem allowed the use of early computational facilities. Moreover, quantum chemistry codes could benefit directly from the fast growing experience in the implementation of linear algebra methods. Boys introduced Gaussian type orbital (GTO) functions for the calculation of molecular integrals [16]. With these functions a general strategy for the calculation of molecular integrals could be developed. In terms of GTO functions, all integrals for the Hartree – Fock energy calculation can either be solved analytically or reduced to the incomplete gamma function. Thus, with the introduction of GTOs the task of solving complicated multi-center molecular integrals was reformulated into the problem of solving huge numbers of one-center Gaussian integrals. This represented a considerable simplification and opened the door to more systematic approaches. Soon it was realized that recurrence relations can be of great help to treat the huge number of integrals efficiently. However, many years were still to come before the first general integral algorithms for GTOs appeared on the scene. Over the years the linear combination of Gaussian type orbitals (LCGTO) approximation became a standard for ab initio Hartree – Fock methods. On the other hand, modern density functional theory (DFT) started with the famous 1964 paper by Hohenberg and Kohn [17], followed by the method of implementation by Kohn and Sham (KS) [18]. With these contributions, a new conceptual way of approaching the many-body problem was opened. On the basis of a formally rigorous theory, the electron density of any system was recognized as containing all the necessary information to describe its ground state. In addition, the idea of mapping the exact density of an interacting system to that of a non-interacting model, which is more easily solved, provided an alternative to the conventional wave function approach. At this moment, the so-called statistical methods like Thomas– Fermi, Thomas– Fermi – Dirac, etc. and the KS method merged to open the new avenue of DFT. Thus, 1964 marks the introduction of one of the most important concepts in KS theory: the exchange-correlation energy functional, an unknown and universal quantity which contains all the information about an electronic system. It is also worth mentioning that immediately after the presentation of the HK theorems, several extensions appeared in the literature, like the finite temperature extension of Mermin [19], that opened the door to the DFT description of inhomogeneous fluids in classical statistical mechanics [20]. In relation to the basis

Applied density functional theory and the deMon codes 1964– 2004

1081

sets mentioned in the previous paragraph, for first-principles DFT methods, the form of the exchange-correlation potential resulted in integrals that could not be solved analytically. Therefore, a much larger variety of basis functions was introduced in DFT methods than in Hartree –Fock methods. Thus, the 1960s was a very fruitful decade, with new concepts developed through the HFS and KS approaches. Even if the first KS calculations, within the local density approximation (LDA), appeared in this decade [21], it is only in the 1970s that the systematic development of local correlation functionals as well as applications to a growing number and variety of systems appeared. 38.2 THE 1970s Even though both Hohenberg– Kohn and Kohn –Sham papers have been subsequently recognized as extremely important for Chemistry, that recognition came late in the community of theoretical chemists. Meanwhile, the MS-Xa method received much more attention. For example, in 1970, Johnson and Smith addressed polyatomic molecules such as perchlorate and sulphate ions for the first time [13]. A landmark application of MS-Xa was the first investigation by Johnson and Smith of the electronic structure of a coordination compound, namely the permanganate ion [22]. The interest in the MS-Xa method for calculating the electronic structure of transition metal complexes increased rapidly and realistic results were soon obtained [23 – 25]. This was the starting point for a large range of MS-Xa applications, including valence band and ESCA photoemission spectra of molecules [26– 28], photoelectron spectra [29 – 32], chemisorption and catalysis [24,33,34], geometrical and electronic structures of metal clusters [35 – 37], metal dimers [38 –40], inorganic species [41 –46], and the inclusion of relativistic corrections [47,48] for heavy elements. Further ambitious investigations were devoted to the study of biosystems such as the ferrodoxin active site [49] or porphine systems [50]. Finally, in the late 1970s Karplus and Case developed a general formalism and performed the first one-electron property calculations using the MS-Xa method [51]. Their contributions also constituted a basis for numerous further publications of molecular properties such as hyperfine tensors [52]. Despite its great success in describing one-electron properties of molecules and solids, the MS-Xa method was unreliable in the description of geometries, mainly due to the muffin-tin approximation. The main contribution to further progress was the introduction of LCAO functions, based on GTOs [53,54] or Slater type orbitals (STOs) [55]. Since then, methodological developments have been possible, allowing the computation of properties for large systems both Xa (HFS) and, later, DFT-based methods (for a review of the use of GTOs with Xa or DFT models, see Ref. [56]). Charge density fitting, introduced first by Baerends et al. [55] in the discrete variational method (DVM) [57], allows one to reduce the N4 problem to N3, without losing accuracy. Sambe and Felton [54] proposed to fit the exchange potential also using an entire set of auxiliary functions, in addition to the charge density fitting. The contribution made by Dunlap et al. [58] in improving the fitting procedure yielded more accurate total energies using finite basis sets. In this approach, which has been used in all versions of the DFT program deMon, References pp. 1092– 1097

1082

Chapter 38

which has been developed in our groups, the self-interaction error of the Coulomb energy from the density and an auxiliary function density is variationally minimized. The variational nature of the fit ensures that only second order errors enter the energy expression. Moreover, the approximate energy expression remains variational. Therefore, analytic derivatives can be accurately calculated within this approach. Originally introduced within the Xa method, this approximation was taken over by many LCGTO – DFT implementations. Many years later, the variational fitting was recast into the socalled resolution of the identity by Almlo¨f and co-workers [59]. Based on this work the variational fitting has today entered the Hartree –Fock-based methods, too. It is still a very active research field. These technical aspects originally developed in LCAO-Xa methods have been of great benefit for the later development of DFT codes. It is interesting to note that, for many years, Xa was presented as an independent selfcontained method, and it is only in 1977 that the review of Connolly [26] proposed that it should be viewed as an approximation to the ‘exact’ DFT. In this decade, the formal development of DFT was also an active area of research. One relevant work was the spin DFT extension of von Barth and Hedin, who also presented a local functional [60]. An important contribution to chemistry in this decade came from the Gordon and Kim approach to treat molecular interactions [61]. It was surprising, as well as encouraging, that this non-variational theory where the kinetic and exchangecorrelation energy contributions were described by local density functionals provided results that were very acceptable and computationally affordable. It was almost immediately recognized that the HK universal energy functional was not free of formal difficulties. The most significant was the v-representability problem that was solved by Levy with his constrained-search approach [62]. As it was mentioned before, the roots of DFT a` la HK are in condensed matter physics; thus, it is not surprising that many of the important contributions in the early 1970s came from this branch of Physics, such as the presentation by Gunnarsson and Lundqvist of the adiabatic connection method for obtaining the exchange-correlation energy functional of the KS method [63]. Parallel to these developments in Solid-State Physics, the problem of evaluating the correlation contribution to the energy was a very active field of research in Theoretical Chemistry. Among the many works in this topic, the modelling of the second order radial distribution by Colle and Salvetti deserves a special mention [64]. This work inspired several researchers to propose new exchange-correlation energy functionals. The wave-vector analysis of the exchange-correlation hole paved the way to the development of several functionals [65]. Several local correlation functionals were proposed in the framework of the KS equations for applications to solids [66,67]. Initiated by a calculation on H2 in 1976 [68], there has been a series of impressive Kohn – Sham – type density functional calculations on molecules performed by Gunnarsson, Harris and Jones [69 –75]. These results showed that the KS-LDA method was able to describe molecular bonding reasonably well, in contrast to the well-known non-bonding effects in the Thomas –Fermi theory. The following decade brought improved descriptions of the exchange-correlation energy functional [76,77] and, consequently, better results. It is also pertinent to look a little bit outside the main stream of DFT development and to review other developments in this period that were important for the development of deMon. At the beginning of the 1970s non-linear optimization methods were already

Applied density functional theory and the deMon codes 1964– 2004

1083

thoroughly investigated in applied mathematics. The broader availability of microcomputers inspired interest in numerical studies of such methods. As a result, iterative optimization algorithms with considerably improved numerical stability were developed. For future versions of deMon, and most other quantum chemistry programs, the development of numerically stable quasi-Newton methods [78] was very important. The most popular second derivative update, named after its authors BFGS [79 – 83], was published in 1970. Most modern geometry optimizers in quantum chemical programs rely on the BFGS update in order to avoid the explicit calculation of second derivatives for the geometry optimization. At the same time, and most likely motivated by similar reasons, Lebedev developed a two-dimensional Gauss-type quadrature scheme (Gauss – Markov) for the unit sphere [84 – 88]. These grids exactly integrate real spherical harmonics up to a maximum degree that is used to characterize them. Today, Lebedev grids, as they are named now, are used in most DFT programs for the numerical integration of the exchange-correlation energy and potential. Also in the same time period, one sees the development of the first systematic molecular integral algorithms for LCGTO approximations [89,90]. For the future deMon development the introduction of Hermite– Gaussian functions as basis functions [91,92] is certainly important, too. These early works inspired us much later to use atom-centered Hermite – Gaussian functions for the expansion of the auxiliary density. This results in short and, therefore, very efficient integral recurrence relations for the three-center Coulomb integrals [93]. In the new version of deMon, Hermite – Gaussian functions are also used for the expansion of the Cartesian Gaussian orbital basis. This part of our integral algorithm is closely related to the original formulation of McMurchie and Davidson [90]. However, a much earlier transformation to Cartesian Gaussian functions is used. A good overview of the early systematic integral algorithms for Gaussian type functions is provided by the 1983 review by Saunders [94]. 38.3 THE 1980s At the beginning of this decade, the Xa methods, i.e. without explicit correlation, were still used in Chemistry. MS-Xa was used for the study of large systems, in particular for magnetic properties [95 –101], and LCAO-Xa (also called HFS) for electronic structure investigations of small metallic systems [102 – 106]. A big improvement was reached when the correlation energies of the spin-paired and spin-polarized uniform electron gas, calculated accurately by Ceperley and Alder [76], were incorporated in the LDA correlation functionals. By the early 1980s, the first reviews on DFT appeared [107 –110], underlining its great potential for applications to atoms and molecules. Also the first publications about analytic gradients for the Kohn –Sham method appeared in the literature [111,112]. This clearly moved the Kohn –Sham method into the computational chemistry arena. Still it took almost a decade more before stable geometry optimization with analytic gradients [113] was available within the framework of the Kohn –Sham method. During this decade, electronic structure calculations of metal dimers [114 –117], clusters [118 – 123] and organometallics [124 – 128] led perhaps to the largest success and References pp. 1092– 1097

1084

Chapter 38

progress beyond the traditional ab initio methods, over a broad range of molecular studies. For the first time, the delicate balance between exchange and correlation effects was proved to be responsible for previous failures in the quantitative treatments of metal –metal bonds [114]. From this period, incorporation of explicit correlation in the KS equations became more and more the standard. Despite the overestimation of binding energies, LCAO-LSD calculations were able to provide good geometries, even for metal –metal bonds [115], reasonable ionization potentials [129] and UV spectra [130 – 132]. Hence, LCAO-LSD studies followed naturally the pioneering LCAO-Xa results for a very large variety of molecular applications, including sufficiently large metallic systems and taking into account, when necessary, relativistic and/or spin orbit corrections [133, 134]. Incorporation of model core potentials in LCGTO – LSD [135 – 137] allowed larger metallic systems to be handled (including eventually relativistic effects in an approximate, but very inexpensive, way) and thus further work was possible on chemisorption models on metallic surfaces [138 – 141]. The 1980s also witnessed a continuous development of new exchange and correlation functionals, incorporating the effects of the non-locality of the density. Indeed, quite soon, it was recognized that the local exchange energy term was introducing a substantial error in the total energy. The so-called generalized gradient approximation (GGA) model of Perdew and Wang [142] initiated a long series of gradient corrected functionals, which led to much more accurate energetic properties. The introduction in this later work of the enhancement function that measures the deviations from the electron gas behavior deserves special mention. Plots of these enhancement functions with respect to the exchange dimensionless scale length, allowed a classification of the exchange energy functionals into three categories. First, those that did not fulfill the Lieb –Oxford bound, like Becke [143], second, those that approached this bound asymptotically, like the family later introduced by Perdew et al. [144], and finally, those that either go to zero or to a value smaller than the Lieb – Oxford bound for large density gradients. To this latter class belongs one of the most commonly used exchange functional in deMon, Perdew86 [142]. The implementation in deMon of this energy functionals uses a cut off for large gradients that makes this functional different from the original one [145]. At the end of the 1970s and the beginning of the 1980s attempts to go beyond the LDA were made through the gradient expansion approximation (GEA). The divergence of the functional derivative of the exchange-correlation energy functional was a serious problem. It was necessary to wait until the GGA made its appearance to have an exchange-correlation functional that was really incorporating the inhomogeneities of the electron density. The real-space analysis of the exchange-correlation hole [146] provided working functionals that satisfy many known conditions. Along this line it is important to note that the functional developed by Perdew et al. [147], that is commonly known as PW91, is the GGA functional that satisfies the most conditions. A very important step in our understanding of the exchange-correlation contribution to the energy was made with the series of works published mainly by the New Orleans school, led by Levy and Perdew. In a series of very important papers, these authors presented rigorous scaling and virial conditions that the exact exchange and correlation energy functionals have to

Applied density functional theory and the deMon codes 1964– 2004

1085

satisfy [148 –151]. The discovery of these conditions was used in the development of new functionals. The view at the end of the 1980s was very optimistic. The formation energies reported by Becke with his functional and his basis-set-free program, NuMol, showed that the GGAs were in reality capable of providing results with chemical accuracy. It is commonly said that it was these results that turned the heads of many theoretical chemists to DFT. In the mean time, the North Carolina team, led by Robert Parr made many important contributions to the application of DFT to chemical problems. He focused his attention on using DFT to justify and explain the origin of many chemical concepts, opening what is now sometimes misleadingly called conceptual DFT. Parr’s group also contributed to the practical use of DFT in chemistry with the development of the correlation functional known as LYP that used an analysis of the radial distribution function together with the local thermodynamics approach to model the pair distribution function. To the surprise of some people, and the delight of others, the combination of the GGA from Becke for the exchange and LYP for the correlation, that produces the very well known BLYP method, provided atomization energies that were in very good agreement with the experimental values. The first implementations of the GGA were not self-consistent. The SCF was done within the local approximation and then, perturbatively, the gradient corrections were incorporated to the total energy. In the early 1990s full self-consistency was implemented in several codes, deMon amongst them [145]. The results obtained with the GGAs were certainly encouraging. The achievement of chemical accuracy together with the low computational cost compared with conventional wave function methods that produce the same thermodynamic accuracy was a strong motivation to establish the limits of applicability of these methodologies. Actually, quantitative comparisons will appear in the next decades, after Becke’s proposals of the ‘half-and-half’[152] and the B3LYP hybrid schemes [153]. Finally, in 1985, the technique introduced by Car and Parrinello [154] to minimize simultaneously the electronic and nuclear coordinates, together with the use of a plane wave basis, has been at the origin of the explosion of applications in material sciences in the next decades. Important developments for the LCGTO – DFT methods using auxiliary functions for the fitting of the Coulomb potential during the 1980s were the introduction of (Cartesian) auxiliary function sets with shared exponents [155,136]. With this technique, the computational effort for the calculation of the three-center Coulomb integrals was considerably reduced [156]. In the now increasing number of calculations with LCGTO – DFT type methods using auxiliary functions, self-consistent field (SCF) convergence problems frequently appeared [157]. It took some time before it was realized that the variational fitting of the Coulomb potential not only reduces the scaling of the integral calculation but also influences the SCF procedure. The 1980s also saw an impressive revival of integral algorithm development for the LCGTO approximation. It was initialized by the work of Obara and Saika in 1986 [158]. In fact, the first deMon and DGAUSS versions had explicit integral routines for different shell combinations based on the Obara – Saika algorithm. Even though these routines were quite fast, they were later substituted by recurrence relations in order to allow higher References pp. 1092– 1097

1086

Chapter 38

angular momentum basis and auxiliary functions. The early deMon versions were restricted to d type orbitals and auxiliary functions only. Another important technical contribution to the development of LCGTO – DFT programs was the introduction of the so-called fuzzy Voronoi polyhedra for the calculation of atomic weights in the numerical integration by Becke [159]. Due to its complicated form, the exchange-correlation potential integrals cannot be solved analytically. Therefore, numerical integration schemes are necessary. This was a large technical drawback for early LCGTO –DFT implementations. The numerical integration scheme of Becke solved this problem in a way that was easy to implement. Based on this scheme, very efficient numerical integrators were developed within a decade. Today, the numerical integration of the exchange-correlation potential is a minor problem in LCGTO – DFT codes. Nevertheless, the accuracy of this integration is still a matter of discussion [160,161]. An important technical development for the geometry optimization and transition state search was the investigation of the Levenberg – Marquard algorithm [162,163] by Banerjee et al. [164]. In this work the authors proposed theoretically well-founded step size selections for quasi-Newton methods, a problem particularly important for the transition state search. The algorithm was recast by Baker [165] substituting the quadratic Taylor series expansion by a Pade expansion. The resulting algorithm has entered the literature as (partitioned) rational function optimization (P)-RFO. It is the basic algorithm for the step selection in the geometry optimization and transition state search in the current version of deMon. It has proven very stable, even for large scale optimizations involving hundreds of atoms. The intense efforts of the DFT community during the 1980s, both in numerical and methodological improvements allowed the development of more sophisticated programs, which, in some cases, became available in the early 1990s. 38.4 THE 1990s The first DFT codes appearing in 1990 were DMOL [166,167] and deMon [168,169], followed by DGAUSS [170 – 172] in 1991, ADF [173,174] and Gaussian [175] in 1992. The deMon code was originally published as a new LCGTO – MCP code, based on the same solution of the KS equations as in previous work [54,56], incorporating MCPs, [135 – 137], as well as a new algorithm for geometry optimization. This algorithm, called hybrid, adopted the Car– Parrinello approach of minimizing simultaneously the orbitals and the nuclear coordinates, but adding a few SCF iterations at each geometry to keep the system close to the Born – Oppenheimer surface. It turned out that this approach was not competitive to the traditional quasi-Newton optimizer that was also included in deMon. As already mentioned, the first version of deMon had explicit integral routines for the three-center Coulomb integrals. The geometry optimization worked already with analytic gradients introduced by Fournier et al. [113]. Local and gradient corrected functionals were available. Shortly after its appearance, the original deMon code was substantially modified for commercialization by BIOSYM Technologies. The beta release of this version appeared in 1993. It inspired the deMon-KS1 [176] series of programs developed

Applied density functional theory and the deMon codes 1964– 2004

1087

in Montreal until 1997. In combination with DFT-optimized basis sets and auxiliary functions [177] this deMon version was used for a large number of applications on small systems with up to 20 atoms. It also served as a basis for the implementation of property calculations within the Kohn –Sham method, including the calculation of NMR shielding tensors and spin – spin coupling constants [178 – 184], EPR parameters [185 – 187], nuclear quadrupole coupling constants [188], simulation of photoelectron [189 –192], IR, and Raman spectra [193 – 195], and the calculation of molecular polarizabilities [196,197]. For visualization, interfaces to MOLDEN [198], MOLEKEL [199], and VU [200] were implemented. The topological analysis of molecular fields, primarily the molecular electrostatic potential and the density, was also realized with this deMon version [201]. At the same time, the original deMon version was further developed in Montpellier and in Stockholm. Larger independent developments included the implementation of the orbital symmetry analysis and the calculation of X-ray spectra [202,203]. Both of these developments were first independent from the Montreal version. In 1997, they merged to the deMon-KS3 [176,202 –204] series of programs. The improved implementation in deMon-KS3 allowed calculations of systems with roughly 50 atoms. This version allowed, already in the early 1990s, the study of a broad range of properties. Applications of DFT in various domains of chemistry were thus performed, with particular interests in NMR and EPR properties of bioorganic molecules [205,206], materials [183], transition metal compounds [207 –210], transition metal clusters, naked or interacting with small molecules [138,139,141,211 – 227], Mo¨ssbauer [228], and ZEKE [229] spectroscopies, optical properties with TD –DFT [192,230– 232], and reactivity [233,234]. In the exchange-correlation functional development arena in this decade it is worth mentioning the detailed analysis of the relation between the exchange functional and the kinetic energy functional that was presented by Roy Gordon [235,236]. Since the 1970s it was very well known that the LSDA or GEA energy functionals did not contain the van der Waals contribution. The underbinding of the local and gradient corrected energy functionals when applied to the description of rare gas interactions was well known. But, their behavior in other weak bonding situations was unknown. Thus, establishing the limits of applicability was a very important issue. In the first half of the 1990s deMon was used to test these functionals in the description of hydrogen bonding [237] and charge transfer complexes [238,239]. In the first case, two intermolecular and two intramolecular hydrogen-bonded systems were tested, including the basis set superposition error using the counterpoise method. LDA was found to be seriously deficient, and the non-local corrections provided an encouraging improvement. For the charge transfer complexes the set of electron donor –acceptor systems, formed from ethylene or ammonia interacting with a halogen molecule (C2H4…X2, NH3…X2; X ¼ F, Cl, Br, and I) were tested. Similar to the hydrogen-bonded systems, it was found that the LDA provides a strong overestimation of the intermolecular interaction. The GGA moved the results in the right direction but not nearly enough; large errors remain. Hybrid functionals were tested, and it was found that the parameters related to the intermolecular interaction for the so-called half-and-half potential are in very good agreement with those obtained through second-order Moller – Plesset calculations and with available experimental data. Interestingly, the widely used and well-known three-parameter B3LYP functional does not perform well; the hybrid methods are not a panacea. References pp. 1092– 1097

1088

Chapter 38

These contributions showed that DFT has to be used with caution when the system under study has important weak inter or intra molecular bonds. During this decade it became clear that it was necessary to go beyond the GGA. In deMon this was done with the LAP [240 – 244] and t functionals [245] in which the correlation functional was derived from an explicit integration over the coupling constant in the adiabatic connection approach. Independent of the deMon development, the AllChem project [246] was started in Hannover in 1995. The main aims of this project were the development of recursive integral routines for the three-center Coulomb integrals [156,93], a stable and efficient numerical integrator for the exchange-correlation potential [247] and an improvement in the SCF convergence behavior of LCGTO –DFT programs using the variational approximation of the Coulomb potential. The recursive integral algorithm developed for AllChem was based on the PRISM algorithm of Pople and co-workers [248,249] and took into account the special structure of the auxiliary function sets used in deMon. For the calculation of the incomplete gamma function an algorithm that ensures close to machine precision (16 decimals) was implemented [250]. The main focus in the development of AllChem was numerical stability in order to avoid noise in the SCF procedure. This was believed to be one of the main reasons for the SCF convergence problems of previous deMon versions. With the same aim an adaptive numerical integrator for the exchangecorrelation potential was developed. It was motivated by the work of Perez –Jorda et al. [251]. However, the radial integration was excluded from the adaptive grid construction. This was the key to success in obtaining an efficient and reliable adaptive numerical integrator. Very recently, the adaptive radial integration has again attracted attention [252,253]. The abscissas and weights of the Lebedev grids were recalculated using quadruple precision [250]. The stability of this integrator was extensively tested in numerically sensitive hyperpolarizability calculations [254,255]. In this calculation, it was shown that the adaptive grid automatically adapts to basis sets. For the generated number of grid points this effect can be large. The adaptive grid has also been used for the geometry optimization and frequency analysis. For these calculations, weight derivatives [256] are implemented in deMon. However, our experience has shown that these derivatives are only important if very small adaptive grids are generated. This is different from recent studies with fixed grids [161]. Because very small adaptive grids are not reliable, weight derivatives are not used in the standard setting for the numerical integration in deMon. Nevertheless, they can be activated by a grid keyword option. Despite these efforts to improve the numerical stability of the LCGTO – DFT implementation, only little improvement was seen in the SCF convergence behavior. A major breakthrough for the solution of this problem was the recasting of the SCF procedure into a MinMax problem if the variational fitting of the Coulomb potential is used [257]. Based on this approach efficient convergence accelerators were developed and implemented, first in AllChem and later in new deMon versions. To improve the SCF convergence, a GTO-based DFT tight-binding approach [258] has been implemented in deMon for the generation of start densities. The advantage of this start density generator is that it adapts to the basis set and does not involve any parameterization. Therefore, it can be used for all elements of the periodic table. Very recently, it has been extended for the use of effective core potentials.

Applied density functional theory and the deMon codes 1964– 2004

1089

38.5 THE 2000s In March 2000, the first deMon Developers meeting was held in Ottawa. At this meeting the deMon and AllChem developers agreed to merge their codes in order to keep a Tower of Babel from propagating. As a result, the new code deMon2K couples the deMon functionality with the stable and efficient integral and SCF part from AllChem. Besides the merging of the two codes the new deMon version possesses improved integral recurrence relations [93], also for effective core potential and model core potential integrals, as well as an improved numerical integration scheme [259]. The main difference to the adaptive integrator of AllChem is the grid generating function. In AllChem the diagonal elements of the overlap matrix were used as the grid generating function. Because the convergence of the numerical integration is directly related to the absolute value of the corresponding quantity this approach ensured the convergence of the full overlap matrix and, therefore, the convergence of the numerical integration of the electron density. From the experiences with this approach we learned that the reliable numerical integration of the electron density is not sufficient for the reliable numerical integration of the exchange-correlation potential matrix. Therefore, in deMon2K the diagonal elements of the exchange-correlation potential matrix are used as the grid generating function. Because a density is needed for the calculation of these elements, the adaptive grid is generated twice in deMon2K. First, it is generated with the start density and after the convergence of this SCF the grid is rebuilt using the converged SCF density. With this grid the energy is then converged again. This usually takes only three or four SCF cycles. Because the grid generation is very fast this step does not slow down the overall performance of the program. Besides the implementation of a new grid generating function, the adaptive grid in deMon2K also works with a new cell function that can be screened for large molecules. It has been noticed in the literature [260] that the calculation of the cell function according to Becke [159] scales cubically with respect to the number of atoms. Already for systems with a few hundred atoms this step can become dominant in the calculation. By introducing a cell function that can be screened, this problem is avoided. This also opens the door for a direct grid generation step that would avoid the use of a grid tape. Work in this direction is currently under way in our laboratory. Due to the introduction of Hermite – Gaussian auxiliary functions, asymptotic expansion for the three-center Coulomb integrals could be derived [93]. With this expansion the calculation of the three-center Coulomb integrals scales nearly linearly. By using the approximated density for the calculation of the exchange-correlation energy and potential [261] a very fast construction of the Kohn –Sham matrix is obtained. Therefore, for systems with more than 3000 basis functions, the linear algebra part of the program becomes dominant. In combination with the MinMax SCF procedure, molecules with several thousand basis functions can be routinely calculated. For the optimization of these systems delocalized internal coordinates [262 –264] have been implemented in the new deMon2K version. The optimization of systems with several hundred atoms is now feasible. In fact, the previous time bottleneck has now been shifted to a memory bottleneck. In Fig. 38.1 the requested RAM size for some benchmark systems is depicted. As this figure shows, a system with 5000 –6000 basis functions can still be calculated References pp. 1092– 1097

1090

Chapter 38

Fig. 38.1. Core memory allocation of deMon for SCF calculations with 1000 up to more than 11,000 basis functions.

on a 32-bit architecture because the RAM request is less than 2 GB. For larger systems, like the depicted double unit cell of the ZSM-5 zeolite in the upper right corner, a 64-bit architecture is required. This system possesses around 11,000 basis functions and more than 700 atoms. The improved performance of the new deMon2K version is also very useful for Born – Oppenheimer molecular dynamics (BOMD) simulations. This technique was already introduced into deMon-KS1 in the 1990s [265 – 268]. However, due to the computational limitations of this early deMon version only small systems could be treated. With the new deMon version BOMD simulations may be performed for systems with 50 –100 atoms over several picoseconds [269]. With the incorporation of a QM/MM embedding scheme [270,271] the first step towards multi-scale modeling has been performed in deMon2K. The improved numerical stability of the new deMon2K version also opened the possibility for accurate harmonic Franck –Condon factor calculations. Based on the combination of such calculations with experimental data from pulsed-field ionization zero-electron-kinetic energy (PFI-ZEKE) photoelectron spectroscopy, the ground state structure of V3 could be determined [272]. Very recently, this work has been extended to the simulation of vibrationally resolved negative ion photoelectron spectra [273]. In both works the use of newly developed basis sets for gradient corrected functionals was the key to success for the ground state structure determination. These basis sets have now been developed for all 3d transition metal elements. With the simulation of vibrationally resolved photoelectron spectra of small transition metal clusters reliable structure and

Applied density functional theory and the deMon codes 1964– 2004

1091

electronic state predictions become possible. This work nicely demonstrates how closely applied DFT and experiment can come together. ´ 38.6 RE´SUME In this review we tried to follow the development of applied DFT over the last 40 years from the perspective of those involved in developing the deMon suite of LCGTO – DFT programs and their underlying methodologies. Modern DFT programs are based on the fundamental work of Hohenberg and Kohn that provides a solid theoretical foundation. However, many technical developments originally proposed within the Xa methodology have found their way into these programs, too. With the introduction of LCGTO DFT programs, a bridge between the Hartree – Fock and DFT worlds was built. Both worlds have profited considerably from each other. This process is still under way. The application of DFT methods in chemistry was for a long time, exotic. The Xa method occupied this space for some years. The situation changed rapidly with the introduction of LCAO DFT methods. The possibility of structure optimization brought these methods into mainstream computational chemistry. The late 1980s and early 1990s saw the consolidation of DFT methods in Chemistry. Today, they serve as standard tools for most computational chemists. The development of deMon also demonstrates how different areas impact each other. The wide availability of more and more powerful microcomputers also triggers continuous software development. This is also true for scientific software development. The deMon Developers come from diverse backgrounds and from all parts of the world. They have come together in a loose cooperative consortium that, in our view, represents a situation where the whole is very much greater than the sum of its parts. While the present authors are responsible for errors or omissions in this small review, the credit for the advances is shared with a much wider group cited in the references here and elsewhere. We predict a bright and rich future not only for this group, but for applied DFT in general. 38.7 ACKNOWLEDGEMENTS We would like to acknowledge the following persons for their contributions to the development of deMon. They are, in alphabetical order and with their home countries in parentheses, Yuri Abashkin (Russia), Jan Andzelm (Poland), Alexei Arbuznikov (Russia), Patrizia Calaminici (Italy), Mark Casida (USA), Henry Chermette (France), Steeve Chre´tien (Canada), Clemence Corminboeuf (Switzerland), Claude Daul (France), Helio Duarte (Brazil), Marcin Dulak (Poland), Roberto Flores-Moreni (Me´xico), Elisa Fadda (Italy), Rene´ Fournier (Canada), Gerald Geudtner (Germany), Nathalie Godbout (Canada), Jingang Guan (China), Se´bastien Hamel (Canada), Thomas Heine (Germany), Klaus Hermann (Germany), Christine Jamorski (France), Florian Janetzko (Germany), Martin Kaupp (Germany), Mariusz Klobukowski (Poland), Hisayoshi Kobayashi (Japan), Matthias Krack (Germany), Martin Leboeuf (Canada), Vladimir Malkin (Russia), Olga Malkina (Russia), Gabriel Merino (Me´xico), Tzonka Mineva (Bulgaria), Piotr Mlynarski References pp. 1092– 1097

1092

Chapter 38

(Poland), Seongho Moon (Korea), Benoıˆt Ozell (Canada), Serguei Patchkovskii (Russia), Luca Pedocchi (Italy), Lars G. M. Pettersson (Sweden), Emil Proynov (Bulgaria), Jose´ Ulises Reveles (Me´xico), Eliseo Ruiz (Spain), Nino Russo (Italy), Emilia Sicilia (Italy), Fiona Sim (Scotland), Suzanne Sirois (Canada), Alain St-Amant (Canada), Luciano Triguero (Cuba), Knut Vietze (Germany), Dongqing Wei (China), Tomasz Wesolowski (Poland), Bernd Zimmermann (Germany). 38.8 REFERENCES 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39

P.A.M. Dirac, Proc. Cambridge Phil. Soc., 26 (1930) 376. E. Wigner and F. Seitz, Phys. Rev., 43 (1933) 804. E. Wigner and F. Seitz, Phys. Rev., 46 (1934) 509. E. Fermi, Rend. Accad. Lincei, 6 (1927) 602. L.H. Thomas, Proc. Cambr. Phil. Soc., 23 (1927) 542. E. Fermi, Z. Physik., 48 (1928) 73. E. Fermi, Rend. Accad. Lincei, 7 (1928) 342. J.C. Slater, Phys. Rev., 51 (1937) 846. J.C. Slater, Phys. Rev., 81 (1951) 385. R. Gaspar, Acta Phys. Acad. Scientiarum Hungaricae, 3 (1954) 263. K. Schwarz, Phys. Rev. B, 5 (1972) 2466. K.H. Johnson, J. Chem. Phys., 45 (1966) 3085. K.H. Johnson and F.C. Smith, Phys. Rev. Lett., 24 (1970) 139. G.G. Hall, Proc. Roy. Soc. Ser. A, 205 (1951) 541. C.C.J. Roothaan, Rev. Mod. Phys., 23 (1951) 69. S.F. Boys, Proc. Roy. Soc. Ser. A, 200 (1950) 542. P. Hohenberg and W. Kohn, Phys. Rev., 136 (1964) B864. W. Kohn and L.J. Sham, Phys. Rev., 140 (1965) A1133. N.D. Mermin, Phys. Rev., 137 (1965) 1441. J.S. Rowlinson and B. Widom, Molecular theory of capillarity, Oxford University Press, New York, 1982. B.Y. Tong and L.J. Sham, Phys. Rev., 144 (1966) 1. K.H. Johnson and F.C. Smith, Chem. Phys. Lett., 10 (1971) 219. K.H. Johnson and F.C. Smith, Phys. Rev. B, 5 (1972) 831. K.H. Johnson and R.P. Messmer, J. Vac. Sci. Technol., 11 (1974) 236. R.P. Messmer, C.W. Tucker and K.H. Johnson, Chem. Phys. Lett., 36 (1975) 423. J.W.D. Connolly, Int. J. Quantum Chem., (1972) 201. J.W.D. Connolly, in: G.A. Segal (Ed.), Semiempirical methods of electronic structure; Part A, Plenum, New York, 1977, p. 105. R.P. Messmer, D.R. Salahub and J.W. Davenport, Chem. Phys. Lett., 57 (1978) 29. J. Weber, H. Berthou and C.K. Jorgensen, Chem. Phys. Lett., 45 (1977) 1. D.R. Salahub, J. Chem. Soc., Chem. Commun., (1978) 385. E.L. Anderson, T.P. Fehlner, A.E. Foti and D.R. Salahub, J. Am. Chem. Soc., 102 (1980) 7422. R.P. Messmer, S.H. Lamson and D.R. Salahub, Solid State Commun., 36 (1980) 265. R.P. Messmer and D.R. Salahub, Chem. Phys. Lett., 49 (1977) 59. D.R. Salahub, M. Roche and R.P. Messmer, Phys. Rev. B, 18 (1978) 6495. F.A. Cotton and G.G. Stanley, Inorg. Chem., 16 (1977) 2668. K.H. Johnson, C.Y. Yang, D. Vvedensky, R.P. Messmer, D.R. Salahub, in: W.C. Johnson, J.M. Blakely (Eds.), Interfacial segregation, American Society for Metals, Metals Park, Ohio, 1979, p. 25. D.R. Salahub, L.C. Niem and M. Roche, Surf. Sci., 100 (1980) 199. F.A. Cotton, G.G. Stanley, B.J. Kalbacher, J.C. Green, E. Seddon and M.H. Chisholm, Proc. Natl. Acad. Sci. USA, 74 (1977) 3109. J.G. Norman and D.J. Gmur, J. Am. Chem. Soc., 99 (1977) 1446.

Applied density functional theory and the deMon codes 1964– 2004 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92

1093

D.R. Salahub and R.P. Messmer, Phys. Rev. B, 16 (1977) 2526. J. Weber, Chem. Phys. Lett., 40 (1976) 275. J. Weber, Chem. Phys. Lett., 45 (1977) 261. J. Weber and M. Geoffroy, Theor. Chim. Acta, 43 (1977) 299. A. Goursot, E. Penigault, J. Weber and J.G. Fripiat, New J. Chem., 2 (1978) 469. J. Weber, M. Geoffroy, A. Goursot and E. Penigault, J. Am. Chem. Soc., 100 (1978) 3995. A. Goursot, E. Penigault and J. Weber, New J. Chem., 3 (1979) 675. D.D. Koelling, D.E. Ellis and R.J. Bartlett, J. Chem. Phys., 65 (1976) 3331. J.G. Norman, H.J. Kolari, H.B. Gray and W.C. Trogler, Inorg. Chem., 16 (1977) 987. C.Y. Yang, K.H. Johnson, R.H. Holm and J.G. Norman, J. Am. Chem. Soc., 97 (1975) 6596. H. Sambe and R.H. Felton, Chem. Phys. Lett., 61 (1979) 69. D.A. Case and M. Karplus, Chem. Phys. Lett., 39 (1976) 33. D.A. Case and M. Karplus, J. Am. Chem. Soc., 99 (1977) 6182. H. Sambe, Int. J. Quantum Chem., (1975) 95. H. Sambe and R.H. Felton, J. Chem. Phys., 62 (1975) 1122. E.J. Baerends, D.E. Ellis and P. Ros, Chem. Phys., 2 (1973) 41. B.I. Dunlap and N. Ro¨sch, Adv. Quantum Chem., 21 (1990) 317. D.E. Ellis and G.S. Painter, Computational methods in band theory, Plenum, New York, 1971, pp. 271. B.I. Dunlap, J.W.D. Connolly and J.R. Sabin, J. Chem. Phys., 71 (1979) 3396. O. Vahtras, J. Almlo¨f and M.W. Feyereisen, Chem. Phys. Lett., 213 (1993) 514. U.V. Barth and L. Hedin, J. Phys. C, 5 (1972) 1629. R.G. Gordon and Y.S. Kim, J. Chem. Phys., 56 (1972) 3122. M. Levy, Proc. Natl. Acad. Sci. USA, 76 (1979) 6062. O. Gunnarsson and B.I. Lundqvist, Phys. Rev. B, 13 (1976) 4274. R. Colle and O. Salvetti, Theor. Chim. Acta, 37 (1975) 329. D.C. Langreth and J.P. Perdew, Phys. Rev. B, 15 (1977) 2884. L. Hedin and B.I. Lundqvist, J. Phys. C, 4 (1971) 2064. L. Hedin, B.I. Lundqvist and S. Lundqvist, Solid State Commun., 9 (1971) 537. O. Gunnarsson and B.I. Lundqvist, Phys. Rev. B, 15 (1977) 6006. O. Gunnarsson, J. Harris and R.O. Jones, J. Chem. Phys., 67 (1977) 3970. J. Harris and R.O. Jones, Phys. Rev. A, 18 (1978) 2159. J. Harris and R.O. Jones, J. Chem. Phys., 68 (1978) 3316. J. Harris and R.O. Jones, J. Chem. Phys., 68 (1978) 1190. J. Harris and R.O. Jones, Phys. Rev. A, 19 (1979) 1813. J. Harris and R.O. Jones, J. Chem. Phys., 70 (1979) 830. O. Gunnarsson and R.O. Jones, Phys. Scr., 21 (1980) 394. D.M. Ceperley and B.J. Alder, Phys. Rev. Lett., 45 (1980) 566. R.O. Jones, J. Chem. Phys., 76 (1982) 3098. R. Fletcher, Practical methods of optimization, Wiley, New York, 1988. C.G. Broyden, J. Inst. Maths. Applns., 6 (1970) 76. C.G. Broyden, J. Inst. Maths. Applns., 6 (1970) 222. R. Fletcher, Comput. J., 13 (1970) 317. D. Goldfarb, Math. Comput., 24 (1970) 23. D.F. Shanno, Math. Comput., 24 (1970) 647. V.I. Lebedev, Zh. Vychisl. Mat. Mat. Fiz., 15 (1975) 48. V.I. Lebedev, Zh. Vychisl. Mat. Mat. Fiz., 16 (1976) 293. V.I. Lebedev, Sibirsk. Mat. Zh., 18 (1977) 132. V.I. Lebedev and A.L. Skorokhodov, Russ. Acad. Sci. Dokl. Math., 45 (1992) 587. V.I. Lebedev, Russ. Acad. Sci. Dokl. Math., 50 (1995) 283. M. Dupuis, J. Rys and H.F. King, J. Chem. Phys., 65 (1976) 111. L.E. McMurchie and E.R. Davidson, J. Comput. Phys., 26 (1978) 218. T. Zivkovic and Z.B. Maksic, J. Chem. Phys., 49 (1968) 3083. A. Golebiew and J. Mrozek, Int. J. Quantum Chem., 7 (1973) 623.

References pp. 1092– 1097

1094 93 94 95 96 97 98 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126 127 128 129 130 131 132 133 134 135 136 137 138 139 140 141 142

Chapter 38 A.M. Ko¨ster, J. Chem. Phys., 118 (2003) 9943. V.R. Saunders, in: G.H.F. Diercksen, S. Wilson (Eds.), Methods in computational molecular physics, Reidel, Dordrecht, 1983. C. Daul, C.W. Schlapfer, A. Goursot, E. Penigault and J. Weber, Chem. Phys. Lett., 78 (1981) 304. C.A. Daul and J. Weber, Chem. Phys. Lett., 77 (1981) 593. D.R. Salahub and R.P. Messmer, Surf. Sci., 106 (1981) 415. C.Y. Yang, K.H. Johnson, D.R. Salahub, J. Kaspar and R.P. Messmer, Phys. Rev. B, 24 (1981) 5673. J. Weber, A. Goursot, E. Penigault, J.H. Ammeter and J. Bachmann, J. Am. Chem. Soc., 104 (1982) 1491. J. Kaspar and D.R. Salahub, J. Phys.-Metal Phys., 13 (1983) 311. D.R. Salahub and F. Raatz, Int. J. Quantum Chem., (1984) 173. C. Famiglietti and E.J. Baerends, Chem. Phys., 62 (1981) 407. D. Post and E.J. Baerends, Chem. Phys. Lett., 86 (1982) 176. D.R. Salahub, S.H. Lamson and R.P. Messmer, Chem. Phys. Lett., 85 (1982) 430. A. Pellegatti, B.N. McMaster and D.R. Salahub, Chem. Phys., 75 (1983) 83. M. Morin, A.E. Foti and D.R. Salahub, Can. J. Chem., 63 (1985) 1982. A.K. Rajagopal, Adv. Quantum Chem., 41 (1980) 59. W. Kohn and P. Vashishta, Theory of the inhomogenous electron gas, Plenum, New York, 1983, p. 79. S. Lundqvist and N.H. March, Theory of the inhomogenous electron gas, Plenum, New York, 1983, p. 410. R.G. Parr, Annu. Rev. Phys. Chem., 34 (1983) 631. C. Satoko, Chem. Phys. Lett., 83 (1981) 111. P. Bendt and A. Zunger, Phys. Rev. Lett., 50 (1983) 1684. R. Fournier, J. Andzelm and D.R. Salahub, J. Chem. Phys., 90 (1989) 6371. N.A. Baykara, B.N. McMaster and D.R. Salahub, Mol. Phys., 52 (1984) 891. D.R. Salahub and N.A. Baykara, Surf. Sci., 156 (1985) 605. D.R. Salahub, Adv. Chem. Phys., 69 (1987) 447. T. Ziegler, V. Tschinke and A. Becke, Polyhedron, 6 (1987) 685. B. Delley, D.E. Ellis, A.J. Freeman, E.J. Baerends and D. Post, Phys. Rev. B, 27 (1983) 2132. D. Post and E.J. Baerends, J. Chem. Phys., 78 (1983) 5663. J. Andzelm and D.R. Salahub, Int. J. Quantum Chem., 29 (1986) 1091. R. Fournier and D.R. Salahub, Int. J. Quantum Chem., 29 (1986) 1077. D.R. Salahub, in: J. Davenas, P. Rabette (Eds.), Contribution of clusters physics to materials science and technology, NATO ASI series, Nijhoff, Amsterdam, 1986, p. 143. A. Selmani, J. Andzelm and D.R. Salahub, Int. J. Quantum Chem., 29 (1986) 829. T. Ziegler, J. Am. Chem. Soc., 105 (1983) 7543. T. Ziegler, J. Am. Chem. Soc., 106 (1984) 5901. T. Ziegler, Organometallics, 4 (1985) 675. T. Ziegler, V. Tschinke and C. Ursenbach, J. Am. Chem. Soc., 109 (1987) 4825. A. Peluso, D.R. Salahub and A. Goursot, Inorg. Chem., 29 (1990) 1544. J. Andzelm, N. Russo and D.R. Salahub, Chem. Phys. Lett., 142 (1987) 169. E. Radzio and D.R. Salahub, Int. J. Quantum Chem., 29 (1986) 241. A. Selmani and D.R. Salahub, J. Chem. Phys., 89 (1988) 1529. T. Ziegler, J.K. Nagle, J.G. Snijders and E.J. Baerends, J. Am. Chem. Soc., 111 (1989) 5631. M. Morin, D.R. Salahub, S. Nour, C. Mehadji and H. Chermette, Chem. Phys. Lett., 159 (1989) 472. T. Ziegler, V. Tschinke, E.J. Baerends, J.G. Snijders and W. Ravenek, J. Phys. Chem., 93 (1989) 3050. J. Andzelm, E. Radzio and D.R. Salahub, J. Chem. Phys., 83 (1985) 4573. J. Andzelm, N. Russo and D.R. Salahub, J. Chem. Phys., 87 (1987) 6562. N. Russo, J. Andzelm and D.R. Salahub, Chem. Phys., 114 (1987) 331. J. Andzelm, A. Rochefort, N. Russo and D.R. Salahub, Surf. Sci., 235 (1990) L319. R. Fournier, J. Andzelm, A. Goursot, N. Russo and D.R. Salahub, J. Chem. Phys., 93 (1990) 2919. I. Papai, D.R. Salahub and C. Mijoule, Surf. Sci., 236 (1990) 241. A. Rochefort, J. Andzelm, N. Russo and D.R. Salahub, J. Am. Chem. Soc., 112 (1990) 8239. J.P. Perdew and W. Yue, Phys. Rev. B, 33 (1986) 8800.

Applied density functional theory and the deMon codes 1964– 2004 143 144 145 146 147 148 149 150 151 152 153 154 155 156 157 158 159 160 161 162 163 164 165 166 167 168 169 170 171 172 173 174 175

176

177 178 179 180 181 182 183 184 185 186

1095

A.D. Becke, Phys. Rev. A, 38 (1988) 3098. J.P. Perdew, K. Burke and M. Ernzerhof, Phys. Rev. Lett., 77 (1996) 3865. P. Mlynarski and D.R. Salahub, Phys. Rev. B, 43 (1991) 1399. J.P. Perdew, Phys. Rev. Lett., 55 (1985) 1665. J.P. Perdew, J.A. Chevary, S.H. Vosko, K.A. Jackson, M.R. Pederson, D.J. Singh and C. Fiolhais, Phys. Rev. B, 46 (1992) 6671. M. Levy and J.P. Perdew, Int. J. Quantum Chem., (1985) 743. M. Levy and J.P. Perdew, Phys. Rev. A, 32 (1985) 2010. M. Levy, Int. J. Quantum Chem., (1989) 617. M. Levy and O.Y. Hui, Phys. Rev. A, 42 (1990) 651. A.D. Becke, J. Chem. Phys., 98 (1993) 1372. A.D. Becke, J. Chem. Phys., 98 (1993) 5648. R. Car and M. Parrinello, Phys. Rev. Lett., 60 (1988) 204. J. Andzelm, E. Radzio and D.R. Salahub, J. Comput. Chem., 6 (1985) 520. A.M. Ko¨ster, J. Chem. Phys., 104 (1996) 4114. B.I. Dunlap, Phys. Rev. A, 25 (1982) 2847. S. Obara and A. Saika, J. Chem. Phys., 84 (1986) 3963. A.D. Becke, J. Chem. Phys., 88 (1987) 2547. J.M.L. Martin, C.W. Bauschlicher and A. Ricca, Comput. Phys. Commun., 133 (2001) 189. M. Malagoli and J. Baker, J. Chem. Phys., 119 (2003) 12763. K. Levenberg, Quart. Appl. Math., 2 (1944) 164. D.W. Marquardt, J. Soc. Ind. App. Math., 11 (1963) 431. A. Banerjee, N. Adams, J. Simons and R. Shepard, J. Phys. Chem., 89 (1985) 52. J. Baker, J. Comput. Chem., 7 (1986) 385. B. Delley, J. Chem. Phys., 92 (1990) 508. B. Delley, DMOL, Biosym Technologies, San Diego, CA, 1990. A. St-Amant and D.R. Salahub, Chem. Phys. Lett., 169 (1990) 387. A. St-Amant, Universite´ de Montre´al, Montreal, (1991). J. Andzelm, in: J.K. Labanowski, J. Andzelm (Eds.), Density functional methods in chemistry, Springer, New York, 1991. J. Andzelm, DGAUSS, Cray Research, Eagan, Minnesota, 1991. J. Andzelm and E. Wimmer, J. Chem. Phys., 96 (1992) 1280. P.M. Boerrigter, G.T. Velde and E.J. Baerends, Int. J. Quantum Chem., 33 (1988) 87. G.T. Velde, F.M. Bickelhaupt, E.J. Baerends, C.F. Guerra, S.J.A. Van Gisbergen, J.G. Snijders and T. Ziegler, J. Comput. Chem., 22 (2001) 931. M.J.T. Frisch, G.W. Trucks, M. Head-Gordon, P.M.W. Gill, M.W. Wong, J.B. Foresman, B.G. Johnson, H.B. Schlegel, M.A. Robb, E.S. Repiogle, R. Gomperts, J.L. Andres, K. Raghavachari, J.S. Binkley, C. Gonzalez, R.L. Martin, D.J. Fox, D.J. Defrees, J. Baker, J.J.P. Stewart and J.A. Pople, Gaussian 92, Gaussian, Inc., Pittsburgh, PA, 1992. M.E. Casida, C. Daul, A. Goursot, A.M. Ko¨ster, L.G.M. Pettersson, E. Proynov, A. St-Amant, D.R. Salahub, H. Duarte, N. Godbout, J. Guan, C. Jamorski, M. Leboeuf, V. Malkin, O. Malkina, F. Sim, A. Vela, deMon-KS Version 3.4, 4, deMon Software, Montreal, 1996. N. Godbout, D.R. Salahub, J. Andzelm and E. Wimmer, Can. J. Chem., 70 (1992) 560. V.G. Malkin, O.L. Malkina and D.R. Salahub, Chem. Phys. Lett., 204 (1993) 80. V.G. Malkin, O.L. Malkina and D.R. Salahub, Chem. Phys. Lett., 204 (1993) 87. V.G. Malkin, O.L. Malkina, M.E. Casida and D.R. Salahub, J. Am. Chem. Soc., 116 (1994) 5898. V.G. Malkin, O.L. Malkina and D.R. Salahub, Chem. Phys. Lett., 221 (1994) 91. G. Cuevas, E. Juaristi and A. Vela, J. Mol. Struct. (THEOCHEM), 418 (1997) 231. G. Valerio, A. Goursot, R. Vetrivel, O. Malkina, V. Malkin and D.R. Salahub, J. Am. Chem. Soc., 120 (1998) 11426. G. Cuevas, E. Juaristi and A. Vela, J. Phys. Chem. A, 103 (1999) 932. L.A. Eriksson, V.G. Malkin, O.I. Malkina and D.R. Salahub, J. Chem. Phys., 99 (1993) 9756. L.A. Eriksson, V.G. Malkin, O.L. Malkina and D.R. Salahub, Int. J. Quantum Chem., 52 (1994) 879.

References pp. 1092– 1097

1096 187 188 189 190 191 192 193 194 195 196 197 198 199 200 201 202 203 204

205 206 207 208 209 210 211 212 213 214 215 216 217 218 219 220 221 222 223 224 225 226 227 228 229 230 231 232

Chapter 38 L.A. Eriksson, O.L. Malkina, V.G. Malkin and D.R. Salahub, J. Chem. Phys., 100 (1994) 5066. A.M. Ko¨ster, P. Calaminici and N. Russo, Phys. Rev. A, 53 (1996) 3865. V. Russier, D.R. Salahub and C. Mijoule, Phys. Rev. B, 42 (1990) 5046. C. Daul, H.U. Gudel and J. Weber, J. Chem. Phys., 98 (1993) 4023. M.E. Casida, in: D.P. Chong (Ed.), Recent advances in density functional methods, World Scientific, Singapore, 1995. C. Jamorski, M.E. Casida and D.R. Salahub, J. Chem. Phys., 104 (1996) 5134. I. Papai, A. St-Amant and D.R. Salahub, Surf. Sci., 240 (1990) L604. I. Papai, A. St-Amant, J. Ushio and D.R. Salahub, Int. J. Quantum Chem., (1990) 29. A. Stirling, J. Chem. Phys., 104 (1996) 1254. F. Sim, D.R. Salahub and S. Chin, Int. J. Quantum Chem., 43 (1992) 463. J.G. Guan, M.E. Casida, A.M. Ko¨ster and D.R. Salahub, Phys. Rev. B, 52 (1995) 2184. G. Schaftenaar and J.H. Noordik, J. Computer-Aided Mol. Design, 14 (2000) 123. S. Portmann and H.P. Luthi, Chimia, 54 (2000) 766. B. Ozell, R. Camarero, A. Garon and F. Guibault, Finite Elem. Des., 19 (1995) 295. A.M. Ko¨ster, M. Leboeuf, D.R. Salahub, in: J.S. Murray, K. Sen (Eds.), Theoretical and computational chemistry 3, Elsevier, Amsterdam, 1996. L. Triguero and L.G.M. Pettersson, Surf. Sci., 398 (1998) 70. L. Triguero, L.G.M. Pettersson and H. Agren, J. Phys. Chem. A, 102 (1998) 10599. M.E. Casida, C. Daul, A. Goursot, A.M. Ko¨ster, L.G.M. Pettersson, E. Proynov, A. St-Amant, D.R. Salahub, H. Duarte, N. Godbout, J. Guan, , K. Herman, C. Jamovski, M. Leboeuf, V. Malkin, O. Malkina, , M. Nyberg, L. Pedocchi, F. Sim, L. Triguero and A. Vela, deMon Software, deMon Software, Montreal, 2001. V.G. Malkin, O.L. Malkina and D.R. Salahub, J. Am. Chem. Soc., 117 (1995) 3294. T.B. Woolf, V.G. Malkin, O.L. Malkina, D.R. Salahub and B. Roux, Chem. Phys. Lett., 239 (1995) 186. K. Bellafrouh, C. Daul, H.U. Gudel, F. Gilardoni and J. Weber, Theor. Chim. Acta, 91 (1995) 215. M. Kaupp, V.G. Malkin, O.L. Malkina and D.R. Salahub, J. Am. Chem. Soc., 117 (1995) 8492. M. Kaupp, V.G. Malkin, O.L. Malkina and D.R. Salahub, Chem. Phys. Lett., 235 (1995) 382. M. Kaupp, V.G. Malkin, O.L. Malkina and D.R. Salahub, Chem.-Eur. J., 2 (1996) 24. B. Coq, A. Goursot, T. Tazi, F. Figueras and D.R. Salahub, J. Am. Chem. Soc., 113 (1991) 1485. P. Mlynarski and D.R. Salahub, J. Chem. Phys., 95 (1991) 6050. A. Goursot, I. Papai and D.R. Salahub, J. Am. Chem. Soc., 114 (1992) 7452. M. Castro and D.R. Salahub, Phys. Rev. B, 47 (1993) 10955. L. Goodwin and D.R. Salahub, Phys. Rev. A, 47 (1993) R774. A. Goursot, I. Papai and D.R. Salahub, Studies Surf, Sci. Catal., 75 (1993) 1547. M. Castro and D.R. Salahub, Phys. Rev. B, 49 (1994) 11842. M. Castro, D.R. Salahub and R. Fournier, J. Chem. Phys., 100 (1994) 8233. A. Martinez, A. Vela, D.R. Salahub, P. Calaminici and N. Russo, J. Chem. Phys., 101 (1994) 10677. P. Calaminici, A.M. Ko¨ster, N. Russo and D.R. Salahub, J. Chem. Phys., 105 (1996) 9546. C. Blanchet, H.A. Duarte and D.R. Salahub, J. Chem. Phys., 106 (1997) 8778. M. Castro, C. Jamorski and D.R. Salahub, Chem. Phys. Lett., 271 (1997) 133. H.A. Duarte and D.R. Salahub, J. Phys. Chem. B, 101 (1997) 7464. C. Jamorski, A. Martinez, M. Castro and D.R. Salahub, Phys. Rev. B, 55 (1997) 10905. A. Martinez, A.M. Ko¨ster and D.R. Salahub, J. Phys. Chem. A, 101 (1997) 1532. D.S. Yang, M.Z. Zgierski, A. Berces, P.A. Hackett, A. Martinez and D.R. Salahub, Chem. Phys. Lett., 277 (1997) 71. A. Martinez, C. Jamorski, G. Medina and D.R. Salahub, J. Phys. Chem. A, 102 (1998) 4643. L.A. Eriksson, O.L. Malkina, V.G. Malkin and D.R. Salahub, Int. J. Quantum Chem., 63 (1997) 575. D.S. Yang, M.Z. Zgierski, A. Berces, P.A. Hackett, P.N. Roy, A. Martinez, T. Carrington, D.R. Salahub, R. Fournier, T. Pang and C.F. Chen, J. Chem. Phys., 105 (1996) 10663. M.E. Casida, K.C. Casida and D.R. Salahub, Int. J. Quantum Chem., 70 (1998) 933. M.E. Casida, C. Jamorski, K.C. Casida and D.R. Salahub, J. Chem. Phys., 108 (1998) 4439. M.E. Casida and D.R. Salahub, J. Chem. Phys., 113 (2000) 8918.

Applied density functional theory and the deMon codes 1964– 2004 233 234 235 236 237 238 239 240 241 242 243 244 245 246 247 248 249 250 251 252 253 254 255 256 257 258 259 260 261 262 263 264 265 266 267 268 269 270 271 272 273

1097

A. Rochefort, P.H. McBreen and D.R. Salahub, Surf. Sci., 347 (1996) 11. H. Guo and D.R. Salahub, Angew. Chem. Int. Ed., 37 (1998) 2985. D.J. Lacks and R.G. Gordon, Phys. Rev. A, 47 (1993) 4681. D.J. Lacks and R.G. Gordon, J. Chem. Phys., 100 (1994) 4446. F. Sim, A. St-Amant, I. Papai and D.R. Salahub, J. Am. Chem. Soc., 114 (1992) 4391. E. Ruiz, D.R. Salahub and A. Vela, J. Am. Chem. Soc., 117 (1995) 1141. E. Ruiz, D.R. Salahub and A. Vela, J. Phys. Chem., 100 (1996) 12265. E.I. Proynov, A. Vela and D.R. Salahub, Chem. Phys. Lett., 230 (1994) 419. E.I. Proynov, E. Ruiz, A. Vela and D.R. Salahub, Int. J. Quantum Chem., (1995) 61. E.I. Proynov, S. Sirois and D.R. Salahub, Int. J. Quantum Chem., 64 (1997) 427. S. Sirois, E.I. Proynov, D.T. Nguyen and D.R. Salahub, J. Chem. Phys., 107 (1997) 6770. H.A. Duarte, E. Proynov and D.R. Salahub, J. Chem. Phys., 109 (1998) 26. E. Proynov, H. Chermette and D.R. Salahub, J. Chem. Phys., 113 (2000) 10013. A.M. Ko¨ster, M. Krack, M. Leboeuf and B. Zimmermann, AllChem, Universita¨t Hannover, 1998. M. Krack and A.M. Ko¨ster, J. Chem. Phys., 108 (1998) 3226. M. Head-Gordon and J.A. Pople, J. Chem. Phys., 89 (1988) 5777. B.G. Johnson, P.M.W. Gill, J.A. Pople and D.J. Fox, Chem. Phys. Lett., 206 (1993) 239. A.M. Ko¨ster: Habilitation thesis, Universita¨t Hannover, Hannover, 1998. J.M. Perez-Jorda, A. Becke and E. San-Fabian, J. Chem. Phys., 100 (1994) 6520. R. Lindh, P.A. Malmqvist and L. Gagliardi, Theor. Chem. Acc., 106 (2001) 178. P.M.W. Gill and S.H. Chien, J. Comput. Chem., 24 (2003) 732. P. Calaminici, K. Jug and A.M. Ko¨ster, J. Chem. Phys., 109 (1998) 7756. P. Calaminici, A.M. Ko¨ster, A. Vela and K. Jug, J. Chem. Phys., 113 (2000) 2199. J. Baker, J. Andzelm, A. Scheiner and B. Delley, J. Chem. Phys., 101 (1994) 8894. A.M. Ko¨ster, P. Calaminici, Z. Gomez, J.U. Reveles, in: G. Parr, K. Sen (Eds.), Reviews of modern quantum chemistry, a celebration of the contribution of Robert, World Scientific, Singapore, 2002. G. Seifert, D. Proezag and T. Frauenheim, Int. J. Quantum Chem., 58 (1996) 185. A.M. Ko¨ster, R. Flores-Moreno and J.U. Reveles, J. Chem. Phys., 121 (2004) 681. R.E. Stratmann, G.E. Scuseria and M.J. Frisch, Chem. Phys. Lett., 257 (1996) 213. A.M. Ko¨ster, J.U. Reveles and J.M. del Campo, J. Chem. Phys., 121 (2004) 3417. J. Baker, A. Kessi and B. Delley, J. Chem. Phys., 105 (1996) 192. ¨ . Farkas and H.B. Schlegel, J. Chem. Phys., 111 (1999) 10806. O J.U. Reveles and A.M. Ko¨ster, J. Comput. Chem., 25 (2004) 1109. D.Q. Wei and D.R. Salahub, J. Chem. Phys., 101 (1994) 7633. D.Q. Wei and D.R. Salahub, Chem. Phys. Lett., 224 (1994) 291. D.Q. Wei and D.R. Salahub, J. Chem. Phys., 106 (1997) 6086. D.Q. Wei, E.I. Proynov, A. Milet and D.R. Salahub, J. Phys. Chem. A, 104 (2000) 2384. S. Krishnamurty, T. Heine and A. Goursot, J. Phys. Chem. B, 104 (2003) 5728. S. Moon, S. Patchkovskii and D.R. Salahub, J. Mol. Struct. (THEOCHEM), 632 (2003) 287. N. Gresh, S.A. Kafafi, J.F. Truchon and D.R. Salahub, J. Comput. Chem., 25 (2004) 823. P. Calaminici, A.M. Ko¨ster, T. Carrington, P.N. Roy, N. Russo and D.R. Salahub, J. Chem. Phys., 114 (2001) 4036. P. Calaminici, A.M. Ko¨ster and D.R. Salahub, J. Chem. Phys., 118 (2003) 4913.

References pp. 1092– 1097