# Molecular Structures from Density Functional Calculations with Simulated Annealing.

код для вставкиСкачатьMolecular Structures from Density Functional Calculations with Simulated Annealing By Robert 0. Jones* The geometrical structure of any aggregate of atoms is one of its basic properties and, in principle, straightforward to predict. One chooses a structure, determines the total energy E of the system of electrons and ions, and repeats the calculation for all possible geometries. The ground state structure is that with the lowest energy. A quantum mechanical calculation of the exact wave function Y would lead to the total energy, but this is practicable only in very small molecules. Furthermore, the number of local minima in the energy surface increases dramatically with increasing molecular size. While traditional ab initio methods have had many impressive successes, these difficulties have meant that they have focused on systems with relatively few local minima, or have used experiment or experience to limit the range of geometries studied. On the other hand, calculations for much larger molecules and extended systems are often forced to use simplifying assumptions about the interatomic forces that limit their predictive capability. The approach described here avoids both of these extremes : Total energies of predictive value are calculated without using semi-empirical force laws, and the problem of multiple minima in the energy surface is addressed. The density functional formalism, with a local density approximation for the exchange-correlation energy, allows one to calculate the total energy for a given geometry in an efficient, if approximate, manner. Calculations for heavier elements are not significantly more difficult than for those in the first row and provide an ideal way to study bonding trends. When coupled with finite-temperature molecular dynamics, this formalism can avoid many of the energetically unfavorable minima in the energy surface. We show here that the method leads to surprising and exciting results. 1. Introduction A knowledge of the exact wave function Y of an interacting system of electrons and ions would allow one to determine many quantities of interest, including the total energy E (Eq. l), where 2 is the Hamiltonian of the system. Calcu- lations of this type can be carried out for systems with very few electrons, the most familiar example being the hydrogen atom. However, approximations for Y are generally unavoidable, and are usually based on the variational principle of RayLeigh and Ritz: If I@) is an approximate wave function, then Equation 2 is valid, where E,, is the exact energy of the ground state. The simplest form of a many-particle wave function is that of Hurtree (1928), who represented @ as a product of single-electron functions (Eq. 3), where each di satisfies a (3) Schrodinger equation whose potential term is given by the mean field of all other electrons (Eq. 4). V,,, is the external (4) [*I Dr R. 0.Jones Institut fur Festkorperforschung Forschungszentrum Julich, W-5170Julich (FRG) 630 0 VCH Verlagsgeselisshafr mbH. W-6940 Weinheim. 1991 field of the nuclei, and the Coulomb potential cp is determined by solving Poisson’s equation (we use atomic units, i.e., the unit of charge is the electron charge e, of distance the Bohr radius uorand of mass the electron mass m), Replacing the product by a single (Slater) determinant-the HartreeFock (HF) approximation-leads to an additional “exchange” potential, without changing the single-particle picture. Although this has been the basic method of atomic and molecular physics for many years, a single determinant does not generally lead to a satisfactory value of the total energy. While a linear combination of determinants (“configuration interaction”, CI) should lead in principle to an exact wave function and energy, the numerical effort required increases explosively with increasing electron number. These arguments apply to a one geometry, that is, for a particular set of internal coordinates of a system. If we have N atoms in an aggregate ( N > 2), the total number of internal coordinates is 3N - 6, since the internal energy of the aggregate is independent of any translational or rotational motion. With so many independent interatomic degrees of freedom, the scale of the problem is clear: 1) Finding the energy minimum by calculating the exact energy for all possible geometries is impracticable, since it would require a vast number of calculations in a system where one is difficult! 2 ) The ability to locate the closest minimum to a particular geometry would simplify the calculations greatly, particularly if the number of possible isomers is small. In general, however, it is difficult to predict the most stable structure, since the number of local minima in the energy surface can be very large indeed. The enumeration of topologically different isomers consistent with a given chemical formula is one of the oldest problems in theoretical chemistry. In 1874, only a decade after the 0570-083319lj0606-0630$3.50+ 2510 Angew. Chem. Inf. Ed. Engl. 30 (1991) 630-640 chemical bond was first denoted by a line joining the nuclei, Cayle-v[’*showed that the number of isomers could grow rapidly with increasing N , and much work in the intervening period has confirmed this. Hoare and McInnes,[zl for example, located all the minima in small aggregates with simple, pairwise interatomic interactions, and found a rapid, perhaps exponential, increase in their number with increasing N . In fact, for aggregates of identical atoms interacting with a pairwise potential, Wille and Vennikt3]have shown that there is no known algorithm for determining the ground state energy and structure that grows with time as a power of N . Such problems are classified as “NP-hard”[41 and are termed intractable. It is indeed sobering when mathematicians derive such a result. Moreover, the parameters in force laws such as those used by Hoare and Mclnnes are usually derived from experiment, and one cannot be sure that the results are not affected by the choice of parameters. This discussion shows that there are two distinct problems to be solved, and the difficulties are multiplicative in nature. The approach we now describe addresses both aspects. First, the density functional formalism provides a numerically eficient scheme for calculating the ground state energy of the interacting system of ions and electrons without attempting to calculate the wave function Y . Second, we use the strategy of simulated annealing (SA) to determine low-lying energy minima. We show that, even in small molecules, there are energetically favorable structures that had not been found on the basis of experiment or intuition. In Section 2 we outline the density functional (DF) formalism and discuss the form of the exchange<orrelation energy Ex,and local density approximations to it. The results of local spin density (LSD) calculations given in Section 3 illustrate the level of accuracy to be expected, and we show in Section 4 how DF calculations can be combined with molecular dynamics. We present some applications in Section 5 and our concluding remarks in Section 6. 2. Density Functional Formalism 2.1. Formal Basis The basic theorems of the density functional formalism ’ ’ showed that: were derived by Hohenberg and K ~ h n . ~They 1) Ground state (GS) properties of a system of electrons and ions in an external field Vex,can be determined from a knowledge of the electron density n(r)alone. The total energy E is such a functional of the density, that is, E = E(n). 2) E(n) satisfies the variational principle E(n) 2 EGs,and the density for which the equality holds is the ground state density nGs. A simple and general proof of these assertions has been given by Levy.[61 The step to a practical scheme for energy calculations resulted from the observation by Kohn and Sham[’] that the minimization of E(n) can be simplified if we write E as in Equation 5. Here To is the kinetic energy that a system with density n would have in the absence of electron-electron interactions, q ( r ) is the Coulomb potential, and the remainder Ex, is our definition of the exchangesorrelation energy. The variational principle now yields Equation 6, where p is the Lagrange multiplier associated with the requirement of constant particle number. Equation 6 applies to the interacting system of electrons and ions. The solution of the corresponding equation for a system of noninteracting particles, Equation 7, can be found 6E(n) 6To 6n(r) - &(r) + V ( r )= p (7) by solving the Schrodinger equation, Equation 8a. The density is given by Equation 8 b, where the sum runs over orbitals i with occupation numbersJ;. The problems (6) and (7) are mathematically identical, provided that V(r) is defined according to Equation 9. This condition can be satisfied in practice by a “self-consistency’’ cycle: From a starting value of n we compute V(r) and the n’ that results, and continue this procedure until n = n’. (9) Robert 0 .Jones was born in 1941 in Kojonup, Western Australia. He graduated in 1963from the University of Western Australia, Perth, and obtained his Ph.D. in 1967 from the University of Cambridge, England, under K Heine. From 1967 to 1970 he worked as Research Associate, Laboratory of Atomic and Solid State Physics, Cornell University, Ithaca, N X USA and since 1970 as Scientist at the Forschungszentrum Jiilich (KFA). In 1975 he was NORDITA Professor at the Chalmers Technical University, Goteborg, Sweden and has been Fellow of the Australian Institute of Physics since 1978. Angen. Chem. lnr. Ed. Engl. 30 (1991) 630-640 631 The problem of determining the total energy of a system of electrons and ions has therefore been reduced to the solution of a single-particle equation of Hartree form (cf. Eq. 4), leading to the energy and electron density of the lowest state and all quantities derivable from them. This method can be extended to the lowest energy state of a given symmetry by restricting the variation to densities compatible with the corresponding symmetry quantum numbers.['] In the form of Equations 8 and 9, density functional theory is an orbital theory and it shares the main advantage of the Hartree-Fock method, namely the interpretation of the results in a singleparticle picture. In contrast to the H F potential, however, the effective potential Y(r) has a local dependence on the density. This situation may seem to be almost too good to be true. Before celebrating, however, we should note the following: 1) The total energy E of the most stable state of a given symmetry is found by solving equations describing a fictitious system of noninteracting "electrons", and the eigenvalues ci and eigenfunctions ll/i in Equation 8 can not be viewed as excitation energies and wave functions of the interacting system. The use of such fictitious systems leads to additional complications, since it is not always possible to draw an unambiguous connection between the states of the interacting and noninteracting systems. In the latter, the f ; will be compatible with a single determinant-or at most a small number of determinants-of the same symmetry.[" 2) While in the energy expression ( 5 ) To, the electron-ion and electron4ectron interaction energies can readily be evaluated, Ex, is dtlfined only as the difference between the sum of these terms and the exact energy. It is a small but far from negligible part of the total energy of atoms and molecules. Moreover, if there is no simple expression for Ex,,how do we determine in practice its functional derivative in Equation 6? Approximations to Ex, therefore play an essential part in applications of D F theory, and we now study its form in more detail. 2.2. Exchange-Correlation Energy The connection between the noninteracting and interacting systems-both of density n ( r + c a n be drawn simply by considering an electron-electron interaction 14 and increasing i. slowly from 0 to 1. One can use this construction to derive an exact expression (10) for Ex,,['] in which nxc is defined as in Equation l l . The pair-correlation function g(r, J , A ) describes the fact that an electron at point r reduces the probability of finding one at J, so that Ex,can be viewed as the interaction between n(r) and the exchange-correlation hole n,,(r). This picture is, of course, very similar to that of the exchange hole introduced by Slarer for electrons of parallel spin."'] nxc(r,J - r) = n(r') i dl(g(r, r', A) - 1) 1 1 Ex, = 2 drn(r) $ dR RZR - 1dSZn,,(r, R ) nxe(r,R), so that approximations for it can give an accurate value even if the description of the nonspherical parts of n,, is very poor. Furthermore, a sum rule following from the definition of the pair-correlation function requires that the exchange-correlation hole contain one electron, that is, for all r , Equation 13 is valid. If we consider - nxc(r,r' - r) as J dJn,,(r, J - r) = - I (13) a normalized weight factor and define locally the radius of the exchange-correlation hole (Eq. 14), Equation 15 follows. Provided the sum rule (13) is satisfied, Ex,depends only weakly on the details of nxcand is determined by the first moment of a function whose second moment is known exactly. The experience of the last twenty years shows that the exact exchange-correlation energy functional E,,(n(r)) must be extremely complicated. The calculation of correlation functions (see Eq. 11, for example) generally requires the knowledge of the exact wave function. Nevertheless, the above argument shows that simple, useful approximations to Ex,may exist. The practical necessity of approximating Ex,is an essential difference between the density functional and configuration interaction approaches. While the latter seeks an exact numerical solution of the Schrodinger equation and precise values of related quantities, even the best solution of the DF equations can only reflect the accuracy of the approximation used for Exc. Furthermore, while the CI approach can be improved systematically by increasing the number of determinants and basis functions, a corresponding scheme for improving the accuracy of LD calculations is presently not known. 2.3. Local Density Approximations for Ex, Local density calculations have a long history in condensed matter physics, where a special role has been played by the approximate exchange potential introduced by Slater.["l It is sometimes overlooked that Slater's derivation was based on the concept of the exchange-hole discussed above, and not on the particular form of the exchange energy density obtained for a homogeneous electron gas. The approximation for Ex,in most widespread use today is the local spin density (LSD) approximation (16). Here c,,(n,, n i ) is (1 1) The isotropic nature of the Coulomb interaction suggests the variable substitution R = J - r in (lo), leading to 632 Equation 12.['l Ex,depends only on the spherical average of the exchange and correlation energy per particle of a homogeneous, spin-polarized electron gas with spin-up and spinAngew. Chem. Int. Ed. Engl. 30 (1991) 630-640 down densities n , and n , , respectively. This approximation is free of adjustable parameters-&,, is taken from the best available calculations for the uniform electron gas-and its use can be justified in systems with small departures from homogeneity. Density distributions are far from homogeneous in atoms, molecules, and most solids, and the usefulness of the approximation relies on arguments such as those in Section 2.2. The approximation satisfies the sum rule (13); more details are given in the review article by Jones and Gunnarsson." '1 small for H, and Li,. In cases where it leads to an energy minimum, the H F approximation usually leads to a small underestimate of the equilibrium interatomic separations. The values for r, calculated using the LSD approximations are generally in good agreement with experiment. An overestimate of 1 - 2 % is common.['5. l61 Ground state vibration frequencies are correspondingly low. The dipole moment of CO and its variation with internuclear separation are given significantly better by the LSD approximation than by H F calculations.[' 7 J 3. DF Calculations with Local Density Approximations 3.2. Alkaline Earth Dimers Be2, Mg,, Ca, Density functional calculations with the LSD approximation are common in extended systems, and the results show some general features. Equilibrium geometries are given reliably in most materials, and energy variations in the neighborhood of local minima are described well, so that vibration (phonon) frequencies are generally in reasonable agreement with experiment. In this section, we select examples from the applications to molecular systems[' 'I to indicate the level of accuracy to be expected in such systems. They illustrate two strengths of the method: the ability to study molecules containing elements from the whole periodic table, and the advantages of an interpretation based on a single-particle picture. They also demonstrate the tendency of LD calculations to overestimate strengths of bonds involving sp orbitals. 3.1. First-row Diatomic Molecules Diatomic molecules of first-row atoms Li-F have served as a testing ground for numerous methods of electronic structure calculation. In Table I we compare measured potential well depths['2] in first-row dimers (homonuclear di- ... While the LSD and H F approximations generally lead to similar bond lengths in first-row molecules, Be, is quite different. The lowest-lying state in the alkaline earth dimers [I a,(fl> 1uu(lJ)]) has equal occupancies of bonding and antibonding orbitals, and Hartree-Fock calculations['81 lead to repulsive curves. Bonds in these molecules have often been referred to as Van der Waals bonds, in the expectation that the bond strength increases monotonically Be, + Mg, -+ Ca, . . . with increasing atomic polarizabilities. The earliest CI calculations supported this picture, since they predicted for Be, a very weak bond with an equilibrium separation near 9 a, .['91 The local density approximation gives qualitatively different results and a different picture of bonding in this homologous series.[201The energy curves show minima in all cases, and the equilibrium separations in Mg, and Ca, agree well with measured values. The bond length in Be, (4.86 a,) was found to be much shorter than in CI calculations. Most striking, however, is the variation in bond energies shown in Figure 1 . The bond in Be, is stronger than in Mg,, and the ('El I\ c\ Table 1. Experimental and calculated well depths [eV] for the experimental ground states of the first-row dimers. Molecule Expt [a] LSD [h] XO [b, cl HF [dl 4.75 1.07 0.10 3.09 6.32 9.91 5.22 1.66 4.91 1.01 0.50 3.93 7.19 11.34 7.54 3.32 3.59 0.21 0.43 3.79 6.00 9.09 7.01 3.04 3.64 0.17 [el 0.89 0.79 5.20 1.28 - 1.37 [a] For Be, from Ref. [25],otherwise Ref. (121. [b] Ref. [15]. [c] Local density calculations with exchange only. [d] Total energies for experimental geometries from Ref. [13]. [el HF calculations for Be, give a purely repulsive energy curve. atomic molecules) with values calculated using HartreeFock,t'31 and local spin density approximations for Ex, and Ex (exchange only).[7% 14,151 The H F approximation leads to substantial underestimates of the binding energies, particularly for singlet ground states. The LSD values overestimate the stability of these molecules, particularly beyond carbon in this row, although the deviations from experiment are Angew. Chrm. Int. Ed. Engl. 30 (1991) 630-640 1" I nL X l l I l " 0 He Be Mg Ca Sr Ba Ra 'xl Fig. 1. Bond energies EB calculated for the state of alkaline earth dimers (full curve, left scale, Ref. [ZO]) and cohesive energies of bulk materials (broken curve, right scale, ref. [22]). Experimental values Refs. [12] and [21]) are given as crosses. variation with atomic number parallels the irregular behavior observed in the bulk cohesive energies (Fig. 1, experiment: ref. [21]; LD calculations: ref. [22]). The bond energies in the molecules and solids are overestimated by the LD approximation in those cases where experimental values are 633 known. The error is significant in Be,, as we show in Table 2, where we include the results of more recent LD and CI calcul a t i o n ~ [241 ~ ~and * gas-phase experimental result^.[^^^ The D F and experimental equilibrium separations are in very good agreement, and the extensive CI calculations reproduce both the well depth and equilibrium separation satisfactorily. Table 2. Experimental and calculated spectroscopic constants of Be, re Experiment [25] CI [24] CI [23] C1[19] LD-LMTO[b] LD-LCA0[15] [%I 4.658 4.73 5 0.03 4.78 4.9 4.67 4.63 we [cm-'1 D, lev] 223.4 . .. ca. 0.1 1 0.09 k 0.01 0.10[a] 0.04 0.48 0.50 ... ... 360 362 ('zi ). 3.3. Diatomic Molecules of the Fourth Main Group: C, ,Si,, Ge, The absence of core p states, the strong attractive potential experienced by the 2p states, and their relatively compact nature apply to all first-row atoms. As a consequence, their bonding properties differ qualitatively from those of atoms in the remainder of the periodic table. The experimental and the excitation ground state in C, , for example, is energy (T, = 0.09 eV) to the 3n,(Z0,n,3) is very small.I'2] The remarkable ease of the og-+ K, transfer is not found in the heavier dimers of this group; these have '2; (2 0: 1 K:) ground states, excitation energies of 1.0 to 1.5 eV to the state (1 K,"), and substantially weaker n 'xl, 'xi [a] Estimated. [b] R. 0. Jones, unpublished results (1982). To discuss the bonding in this homologous series, it is useful to consider the valence orbitals (Fig. 2) of the atoms. With the exception of He,, there is a substantial overlap between the electron densities on the two atoms, particularly in Be,. Furthermore, the radial extent of the orbitals does not increase smoothly with core size, but shows a "secondary periodicity". For example, the valence s orbital in Mg is not Fig. 3. Radial valence orbital functions for atoms of the fourth main group: a) s functions, b) p functions. rla,l - Fig. 2. Radial valence orbital functions for helium and alkaline earth atoms: a) s functions for the ' S (ns') states, b) p functions for the 3P (ns'np') state. The dashed curve is the 2p function of He (ls2p). The arrows denote half of the calculated equilibrium separation (rJ2) in each dimer [20]. The radial functions of the valence orbitals of these elements (Fig. 3) show a similar behavior to that discussed above. The valence functions of Si and Ge are remarkably -0.251 ,K Si, \ 7-===== much more extended than that in Be. The 2p functions in the Mg core have no repulsive (orthogonalization) effect on the 3s function and are sufficiently extended that the additional core charge is incompletely screened. A similar orbital compression is evident in Sr and Ra, where 3d and 4f functions enter the core for the first time. The unoccupied p orbitals are generally more extended than the s valence functions, but Figure 2 shows that Be is an exception. The similar extent of the 2s and 2p orbitals suggests that sp mixing will be favored. This picture is in accord with the pseudopotential theory of cohesion in the bulk: a stronger sp hybridization (and larger band gap) in Be than in Mg, and increasing sd hybridization with increasing atomic number. It is encouraging that the trends predicted by the DF calculations, particularly the relative strength of the bond in Be,, were confirmed by both exhaustive CI calculations and by experiment. 634 -1.00 - -1'2 -1.50 2 3 L r[ool 5 1 5 6 Fig. 4. Dependence of the self-consistent eigenvalues for 'Egstate ofC,, Si,, and Ge, on the interatomic distance r. The valence eigenvalues for the isolated atoms are given and the arrows on the r axis denote half of the calculated equilibrium separation ( r J 2 ) in each case. Angew. Chern. Inf. Ed. Engl. 30 (f991) 630-640 similar, since the relatively diffuse 3d core density in Ge screens the additional nuclear charge imperfectly. This is reflected in very similar bulk properties, such as ground state geometries, and in the eigenvalues of Si, and Ge, shown in Figure 4. The C, eigenvalues reflect the compact valence p function and show why the multiplet structure is so different. state obtained by placing four electrons in the nu The orbital has a very similar energy to the "C, state obtained by transferring two electrons to the 2 crgorbital. The eigenvalues in Figure 4 suggest that occupancy of the latter orbital will be favored in the heavier dimers, resulting in weaker n bonds. The consequences for the formation of simple saturated molecules have been discussed by Harris and Jones.[26J 'El 3.4. Triatomic Molecules of the Sixth Main Group: 03, so, As a further example, we study the energy hypersurfaces of 0, and SO,,[27Jwhich have ground states with 'A,(C,J symmetry and similar bond angles (116.8" and 119.4", respectively). Although the absorption spectrum of ozone has been studied in detail, particularly for ultraviolet radiation, little experimental information is available on the excited state energy surfaces, and theoretical studies have been essential. Attention has been focused in the past on the presence of two minima in the 'A, energy hypersurfaces and the energy difference between them. The H F approximation gives a qualitatively incorrect ordering of the low-lying states of 0,, since there are two configurations with substantial contributions to the ground state wave function. In Figure 5 we show the energy hypersurfaces for lowlying states of ozone. D F calculations reproduce the ground t However, there are qualitative differences between the energy ordering of the states. In particular, the excitation from the ground state to the low-lying triplet states requires substantially more energy for SO, than for O,, and the energy separation between the two 'A, states (4.1 eV) is almost three times as large for SO,. These differences can be related to the larger radius of the sulfur atom and the higher energy of the p 0rbita1.l'~~The ground state (x' A,) has the valence configuration 1a: 3 b: 4a: 2 by, and the 2 'A, state corresponds to the excitation (3 b: -+ 2 b;). While the 1 a, and 3 b, orbitals have significant contributions only from the outer (oxygen) atoms, the higher-lying 4a, and 2b, orbitals have a strong contribution from the central atom (0in O,, S in SO,). The eigenvalue spread in SO, is therefore much larger, so that the large excitation energy xi A, + 2 'A, in SO, reflects simply the double excitation to the high-lying 2b, orbital. The above molecules are only a small sample of those that have been studied with the D F method. Nevertheless, they illustrate some general features of the results. As in extended systems, equilibrium geometries and vibration frequencies are described satisfactorily, even in molecules-such as Be, and 0,-where the Hartree-Fock method gives qualitatively incorrect results. While energy variations near equilibrium geometries are given satisfactorily, this is not true of all regions of configuration space. The errors in the dissociation energies of first-row diatomic molecules and the ozone family (up to ca. 2 eV), for example, are unacceptably large and have contributed to the motivation behind the search for improved approximations.["] The single-particle picture and the relative ease of performing calculations for elements with large atomic numbers is very useful in understanding the differences between first-row and heavier elements. A discussion on similar lines was presented subsequently by Kutzelnigg.[2 4. The Molecular Dynamics/Density Functional Approach EB lev1 9- 1 10 Fig. 5. Energy surfaces for low-lying states of 0,. ec = bond angle. The values on the curves give the bond lengths [a,] that optimize the energy for each angle a. state geometry well, and the excitation energy between the two 'A, minima (1.4 eV) is in reasonable agreement with the most recent CI calculations, which give results between 1.O eV and 1.4 eV.1271In both 0, and SO,, the bond lengths increase in the order x' A, -+ 1 3B, -+ 1 3B, -+ 2 'A,, and the bond angles in the order 2'A, + 1 3B, -+x'A, -+ I3B,. Angen,. Chem. I n [ . Ed. Engl. 30 (1991) 630-640 While we cannot expect to be able to determine the exact ground state if the energy hypersurface has many local minima, it is essential to develop methods for avoiding energetically unfavorable local minima. Kirkpatrick et al.1291noted that this can be achieved by allowing the system to evolve at finite temperature and implemented this "simulated annealing" strategy using a Monte Carlo sampling. Molecular dynamics (MD) provides an alternative, and Car and Parrinelshowed that it could be combined with the density functional (DF) scheme for calculating total energies. The use of the LSD approximation for the exchangexorrelation energy avoids the parameterization of the interatomic forces common in MD schemes,[311and should give a reliable description of energy variations for large regions of configuration space. To perform geometry optimization using D F calculations, we face two minimization problems: The total energy must be minimized for each geometry by varying the density (the solution of Eq. 7), and the geometry with the lowest energy must be found. This is done conveniently by viewing E as a 635 function of two interdependent sets of degrees of freedom, {I)~} and { R , } , and using standard MD techniques (Eq. 17, where 2, are the ionic charges).[301 The energy minimization required for structure determination can be performed using standard MD techniques,[301 since the evolution of the system of electrons and ions is reproduced exactly by the evolution of a dynamic system defined by the Lagrangian (18) and the equation of motion 19a and 19b. M I and R, denote the ionic masses and coordinates, pi are fictitious "masses" associated with the electronic degrees of freedom, dots denote time derivatives, and the Lagrangian multipliers Aij are introduced to satisfy the orthonormality constraints on the tji(r, t ) . From these orbitals and the resultant density n(r, t ) = C I$i(r, t ) l z we 5.1. Sulfur Clusters S, The elements of the sixth main group provide some of the best characterized of small atomic clusters. These elements are unique in that many allotropes comprise regular arrays of well-separated ring molecules, and X-ray structure analyses have been performed for S,, n = 6- 8,lO - 13,18,20, and Sen, n = 6, 8. S, has been found as microcrystals, and S,, exists in two distinct forms. Several mixed crystals of the form Se,S, and a range of sulfur oxides and ions are also known. The preparation and structure of these clusters has been reviewed by S t e ~ d e l . ~ ~ ~ ] The theoretical results for sulfur clusters S , to S, 3[341 show that it is possible to determine low-lying energy minima even if the starting geometries are far from correct: linear chains in S, to S, and nearly planar rings in S, to S 1 3 .The final structures agreed well with experiment in all cases where X-ray data were available. For example, the D,,symmetry in S,, (Fig. 6b) is reproduced very closely, and the structural parameters (bond length d = 3.97a0, bond angle ci = 106", dihedral angle y = 88') are in good agreement with measured values (3.88a0, OL = 106.2", y = 87.2'). The discrepancy in bond length is reduced by using a more accurate pseudopotential, as shown by calculations for the D,, form of S,, where the structure ( d = 3.900a0, LY = 108.5", y = 98.3") is very close to the experimental result ( d = 3.893 0.006a0, ci = 108.0 f 0.7", y = 98.3 f 2.1").[3s] I use Equation 17 to evaluate the total energy E, which acts as the classical potential energy in the Lagrangian function 18. The artificial Newton's dynamics for the electronic degrees of freedom, together with the assumption < M I , prevent transfer of energy from the classical to the quantum degrees of freedom, even for long simulations. 5. Applications of the MD/DF Scheme to Molecules MD/DF calculations have been performed for several families of molecules containing main-group elements, using a large unit cell with periodic boundary conditions to guarantee a weak interaction between the molecules. We give some examples here to show the value of the approach in several contexts.[321The scattering properties of the individual atoms are described by nonlocal pseudopotentials, and the eigenfunctions +i are expanded in a plane wave basis. We set the velocities di and R equal to zero for an arbitrary starting geometry, and use an efficient self-consistent iterative diagonalization technique to determine those iithat minimize E. With the electrons initially in their ground state, the dynamics (Eq. 19a and b) generate Born-Oppenheimer (BO) trajectories without the need for additional diagonalization/self-consistency cycles for the electrons. In the present applications, the minimum of the potential energy surfaces can be found by alternating MD calculations at finite temperature with steepest descents (or conjugate gradient) determinations of the nearest local minimum. Further details are given in the original papers. 636 Fig. 6 . Structures of a) S , and b) S , , Cases where the most stable isomers are not yet known are of particular interest. There is no experimental information on S, and S, ,for example, and S, has only been prepared as crystallites that are too small for X-ray diffraction analysis. The most stable S, isomer in our calculations was an unusual rectangular structure made up of two weakly interacting dimers. This minimum of the energy surfaces is very shallow and it is easy to distort the molecule. The detailed ab initio calculations of Rughavachuri et aI.[361support our assignment of the ground state. This is also true in s,, where we found a ground state isomer with C, symmetry (Fig. 6a) as well as several local minima with C, symmetry with energies > 0.2 eV higher. The calculated ground state structure is consistent with measured Raman spectra, which indicate narrow ranges of bond lengths and bond angles, and dihedral angles in the range 70- 130". 5.2. Structural Change in S,O A surprising example of structural optimization is provided by the S,O molecule.[37]In the course of the simulation Angew. Chem. Inr. Ed. Engl. 30 (1991) 630-64Q shown in Figure 7-1200 time steps at T = 2000K-the molecule changed from one with a stable local minimum and a ring structure similar to that In S, (Fig. 7a) to a structure with different topology: an oxygen atom outside the S, structure (Fig. 79. The structure found on reducing the temperature SlOWlY to T = 0 K (Fig. 7J) agrees well with the experimental ground ~tate.1~’’ The entire structural change I1 21 il 31 I1 I 1 I1 51 Flg 8 Possible cyclic structures for Se,S, and Se,S, The numbers in parentheses indicate the positions of the two atoms of the minority element Calculations have been carried out on seven- and eightmembered rings of the type Se,S,,’471 starting with the ground state structures of S, (C,) or S, (D4J. The eightmembered rings are characterized by small deviations from the crown-shaped (D4J structures, and the possible cyclic structures in SezS6 and Se,S, are shown in Figure 8. The results for these molecules show several remarkable features, illustrated by the structural parameters for Se,S, given in Table 3. In all eight structures, the bond lengths (ds-s,d j _ s , , (el Table 3. Molecular parameters d[u.]. a and y [ “ ] for all isomers of Se2S6(see Fig. 8). Bond lengths between atoms i and j are denoted by d,,,bond angles at atom i by a, and dihedral angles at bond ij by yi,. W Ijl (il Fig. 7. Evolution of S,O from ring (a) to ground state structure 6). The time s. difference between successive frames is 1.1 x (including cooling) took only about 8000 time steps (< 10-’2s), although the energy barrier between ring and ground state structures is ca. 5 eV and the energy difference only about 0.2 eV. This shows just how efficient molecular dynamics can be in generating geometrical configurations. 5.3. Isomers of Se,S,, Se,S, Sulfur and selenium have much in common, and binary systems have been studied extensively in the vapor phaser3*,391 and as liquids and as solid solution^.^^^-^^] The different species tend to crystallize together, leading to structures with sulfur and selenium atoms distributed over the atomic The possible complexity is evident in the eight-membered rings S,Se, -”, where thirty different crown-shaped isomers can occur. Apart from the intrinsic interest in molecules of the form Se,S,, an understanding of their structures could provide insight into the structures of related liquid and amorphous materials, including Se itself. The interconversion of various Se,S, isomers and ringchain equilibration processes involves reactions of the type in (20). This reaction is endothermic in the For -S-S- + -Se-Se- -+ 2 -Se-S- (20) diatomic S-Se molecules in the gas phase,[451the measured enthalpy change is +7.9 f 6.7kJmol-’ and the sum of the dissociation energies on the left side of (20) is greater than the sum on the right. Theoretical predictions of the relative stability of different isomers are difficult to make,’461 and it is natural to ask whether DF calculations can describe reliably the small energy differences involved. Angew. Chem. Inr. Ed. Engl. 30 (1991) 630-640 4.42 4.21 3.92 3.93 3.95 3.93 3.92 4.21 105.9 105.8 108.1 108.8 109.1 109.2 108.8 108.2 96.8 97.2 97.6 100.6 102.9 100.5 97.5 97.3 4.20 4.20 4.20 3.93 3.93 3.93 3.93 4.20 106.1 108.1 106.2 108.4 108.7 109.2 108.7 108.5 98.0 96.6 96.1 100.4 102.1 100.4 99.0 97.6 4.20 3.92 4.21 4.21 3.93 3.92 3.93 4.21 106.7 108.2 108.2 106.7 108.5 108.7 108.8 108.6 98.3 97.8 96.3 99.1 101.1 98.9 99.1 99.1 4.21 3.93 3.93 4.21 4.21 3.92 3.93 4.21 106.7 108.9 108.6 108.4 106.5 109.2 108.7 108.5 99.1 96.6 98.2 100.5 99.0 96.5 97.7 100.4 ds,-s,) are the same to within 0.02 a, (3.93 aor4.21 a,, 4.44a0), and ds-s and dS,+, are the same as found previously for the ring structures of S, and Se,. The (1,2) isomers are the most stable in both Se,S, and Se,S,, and the energies of the other three structures are ca. 0.08 eV higher and degenerate to within 3 meV in each case. We now show that these findings can be understood in terms of a very simple model. The transferability of the bond lengths and the known connection between bond length and bond strengthrz6]suggest that we can associate with each bond type a contribution to the binding energy, E,-,, Es-se, and Es,-s,. This leads to (21) for the (1,2) structure in Se,S6 and to (22) for all remain- 637 ing structures. This argument predicts that the energies of the last three structures will be equal, and will differ from the energy of the (1,2) structure by A E (23). If we apply the same argument to the Se,S, structures, Equations 21 -23 still apply when S and Se are interchanged. This explains why the energy orderings in Se,S, and Se,S, are the same, and why the most stable (1,2) isomers are separated from three structures with almost identical energies. The separation in the present calculations (ca. 0.08 eV, 7.4 kJmol- ’) is consistent with the endothermic reaction (20) and in reasonable agreement with the measured heats of reaction in the gas and liquid phases. In spite of its simplicity, this model-based on a very careful geometry optimization-gives results that agree with trends found in recent measurements, particularly the relative abundance of structures in which the minority atoms are adjacent. The symmetry between the energy ordering of the isomers of Se-rich and S-rich molecules is perturbed, of course, by differences between the elements such as the atomic radius, but the essential features should be unchanged. The trend to segregation of minority components could also occur in the disordered liquid and amorphous states. Segregation in this model is a consequence of the sign of B E in Equation 23, and other systems with a small value of A E or with one of opposite sign would behave differently. 5.4. Phosphorus Clusters P, The structural variety shown by phosphorus exceeds that of all elements other than sulfur and perhaps boron, and there have been many studies of its crystalline and amorphous A problem of continuing interest in gasphase clusters has been the detection of only p,, p,, p,, and (possibly) P, in the vapor phase at high temperatures.[501 There have been numerous speculations about the apparent absence of heavier clusters,[5 and several calculations indicate that the cubic form of P, is less stable than two P, tetrahedra.[52.5 3 1 Recently, Martint541was able to detect by mass spectroscopy P, molecules up to n = 24 in a low-temperature beam. In spite of this, almost nothing is known about the structure of clusters with n > 4. MD/DF calculations have been performed on phosphorus clusters up to PlO,[sJ1 and calculated and experimental geometries and vibration frequencies agree well in those cases (P,, P4) where the latter are known. In the larger clusters there were several unexpected findings: 1) Although the tetrahedral structure is energetically favored in P,, there is a large “basin of attraction” for a D,, “roof” structure, that is, this structure is the closest minimum for a large region of configuration space. 2) The roof structure is a prominent feature in the low-lying isomers in P, to P,. The most stable isomers found in P,, P,, and P, have a P,-roof with an additional one, two, and three atoms, respectively. The two forms of P, shown in Figure 9 are energetically more stable than the cubic structure. The structure in Figure 9 a shows again the preference for the “roof’ structure (two such units are bonded together), and the isomer shown 638 in Figure 9 b was the most stable we found. It can most easily be understood as a (distorted) cube with one bond rotated through 90” and is ca. 40 kcalmol-’ more stable than the cubic form. This “wedge” or “cradle” structure is found as a structural unit in violet (monoclinic, Hittorf) phosphoru~.[~~I ii Fig. 9. Two isomers of P,. There is a pronounced analogy between the structures of the P, isomers and those of the corresponding isoelectronic hydrocarbons C,H,. The cubic form of the latter (cubane) has been prepared by Eaton and and can be converted catalytically to the wedge-shaped form ~uneane.[~’] The third possible isomer of C,H, (analogous to Fig. 9a) was estimated to have a strain energy intermediate between the other two, in agreement with the energy ordering we found in the P, isomers.15s1 The results for the P, clusters show two particularly interesting features: 1) The presence of a large basin of attraction for the energetically unfavorable roof structure in P,. This shows that the most favorable structures are not necessarily surrounded by the most extensive minima in the energy surfaces. 2) The existence of two isomers of P, that are more stable than the cubic form. Although attention had been focused for many years exclusively on the last, our finitetemperature simulation led almost immediately to more stable forms. The lowest energy for P, lies ca. 0.5 eV below the energy of two P, tetrahedra in the LD calculations. It would be interesting to see the energy difference found using ab initio methods. 5.5. Aluminum Clusters Al, The calculations for the sulfur and phosphorus clusters reproduce known structures satisfactorily. This is also true for small aluminum clusters, where spectroscopic data are available for Al,, and ab initio calculations have been performed for a limited range of geometries up to Al, . [ 5 9 1 If we can reproduce known structures of small clusters with our methods, we may have confidence in our predictions for larger clusters of these elements. Aluminum is a prototype “simple metal” of the condensed matter theorist, which is characterized by a weak electron-ion interaction and a valence electron charge distribution that is nearly uniform. It is natural then that calculations of the electronic structure of A1 clusters have often used a “spherical jellium” model,[601 where both the electron density and the positive-ion distribution are assumed to be uniform inside a sphere of appropriate size. The focus in such calculations has been on particularly stable electronic configurations with “magic numbers” of electrons. Angew. Chem. Int. Ed. Engl. 30 (1991) 630-640 MD/DF calculations up to n = 10 give structures for n = 2-4 in good agreement with other calculations, and a satisfactory description of variations in ionization energies. There were, however, some unexpected results for larger clusters,I6’] including the existence of a class of energetically favorable structures that can by no means be viewed as spherical. Figure 10 shows three such structures for Al,. They can be viewed as distorted planar arrays of approximately equilateral triangles, connected so that adjacent triangles are either nearly coplanar (dihedral angle near zero) or have dihedral angles ca. 35“-55”. These features are also apparent in bulk aluminum (the face-centered-cubic structure comprises equilateral triangles with dihedral angles zero or 54.7”) and in the t( form of the neighboring boron-group element gallium, as shown in Figure 10d. There are, of course, many possible structures of this type in molecules of finite extent. It is surprising, nevertheless, that D F calculations show that they are of comparable stability to the more compact “metallic” structures of Al,. The structure shown in Figure 1Oc was, in fact, the most stable found in this molecule. Id (d) Fig. 10. a) -c) Three isomers of Al,. d) Crystal structure of a-Ga. We have noted above how useful the D F approach has been in finding trends in families of molecules, and in relating them to single-particle properties such as the valence orbital. This is also true in the elements of the third main group. We find that the bonds in gallium clusters are generally shorter than the corresponding bonds in aluminum clusters, and that this can again be correlated with the range of the valence orbitals.[611 6. Concluding remarks Theory should be able to contribute much to the determination of molecular structures. The calculation of the energy surfaces and the location of low-lying minima should lead to the most stable isomers, but there are severe problems: The determination of the energy E from the exact wave function is impracticable in all but the smallest molecules, and there are usually many local minima in the energy surface. The approach described here addresses both aspects. The density functional formalism, with the local spin density approximation for the exchange-correlation energy, enables us to carry out energy calculations in an efficient manner, and the use of finite temperature molecular dynamics prevents the molecule becoming trapped in unfavorable energy minima. The results have been encouraging. The structures of small clusters of sulfur are described well and, in the case of Angew. Clirm. Int. Ed. Engl. 30 (1991) 630-640 S,O, we can simulate structural changes involving a high energy barrier. The final structure (an 0 atom outside an S,-ring) is in good agreement with the structure determined by X-ray diffraction. The heterocyclic ring molecules Se,S, are characterized by structures with very small energy differences and energy barriers that are small on a thermal scale. The calculations led in this case to a simple model of the energy differences in mixed Se-S ring structures and predictions for molecules for which calculations have not yet been performed. In the case of phosphorus clusters, we predict structures with plausible trends and with an interesting analogy to isoelectronic hydrocarbons (CH), . In Al, clusters we have found a class of energetically favorable structures that was quite unexpected. There have been so many surprises, in fact, that one can only assume that many calculations using other methods have overlooked important minima in molecular energy surfaces. The number of minima and the fact that many can have comparable stability show how difficult it is to determine structures on the basis of “chemical intuition” alone. There is little doubt that other methods of calculating electronic structures would benefit from the implementation of simulated annealing. The successes of this approach should not lead us to overlook its limitations. Even the most careful calculation reflects the approximations used, and the LSD approximation is an essential part of most of the calculations performed so far. Overestimates in the binding energy are evident in all the molecules discussed above, and there are indications that an inadequate description of exchange energy differences is an important factor.[”] Improved approximations would be useful, if implemented in a systematic way, but attempts in this direction have had mixed success. Moreover, there are many local minima in the energy surfaces even for clusters containing less than 10 atoms. The implementation of molecular dynamics techniques does not change this, and no method can guarantee finding the global energy minimum, or even all of the most important minima. The development of alternative optimization schemes and of simplified energy functionals[621will be important in the future. Density functional calculations are common in condensed matter physics and it is not surprising that workers from this field have played the major role in theoretical developments and in molecular applications. In fact, density functional methods have been slow to find favor amongst chemists. The coupling to molecular dynamics has opened up perspectives not currently available to some of the traditional methods of theoretical chemistry, and-perhaps in conjunction with these methods-the prospects of applications to many problems of chemical interest are very good. I have benefited greatly from collaboration over the years with 0. Gunnarsson, J. Harris, and D. 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Phys. 86 (1987) 7054; L. G. M. Petterson, C. W. Bauschlicher, Jr., T. Halicioglu, J. Chem. Phys 87 (1987) 2205, and references cited therein. [60] For a review, see W. A. deHeer, W. D. Knight, M. Y. Chou, M. L. Cohen, Solid State Phys. 40 (1987) 94. [61] R. 0. Jones, unpublished results. 1621 See, for example, J. Harris, D. Hohl, J Phys. Condens. Matter 2 (1990) 5161, and references cited therein. Angew. Chem. Int. Ed. Engl. 30 (1991) 630-640

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