Ab initio molecular dynamics with correlated molecular ... · Ab initio molecular dynamics with...

Ab initio molecular dynamics with correlated molecular wave functions: Generalized valence bond molecular dynamics and simulated annealing

Bernd Hartke and Emily A. Carter University of California, Los Angeles, Department of Chemistry and Biochemistty, 405 HiIgard Avenue, Los Angeles, California 90024-1569

(Received 20 May 1992; accepted 14 July 1992)

We present an ab initio molecular dynamics algorithm at the generalized valence bond level. It does not need a precalculated potential energy surface or model Hamiltonian; instead the nuclei move according to first principles forces derived from the electronic wave function which in turn follows the movement of the nuclei. This technique includes the dominant static electron correlations, it can treat ground and excited many-electron states, and it can describe chemical bond formation and breaking qualitatively correctly. We apply the method to Na,, as a generic test example for small metal clusters, and show spin- dependent free dissociation dynamics as well as geometry optimization by simulated annealing. The latter involves novel boundary conditions to prevent dissociation and mass scaling to enhance performance.

1. INTRODUCTION

Ab iniriu molecular dynamics ( AIMD), i.e., molecular dynamics without a priori knowledge of the potential energy surface, has several features which make it highly desirable. For example, it imposes no model-induced bias upon the system, like reduced dimensionality Hamilto- nians or truncated many-body potentials. Furthermore, the only parts of the potential energy surface that are calculated are those actually visited in a particular application; this becomes increasingly important with more compli- cated potential energy surfaces and larger systems. Unfor- tunately, the cost of calculating a point of the potential energy surface and/or the corresponding ab initio force on the nuclei makes the method computationally so expensive that its advantages cannot be exploited even for small systems.

This problem was alleviated by the idea to abandon the full recalculation of the wave function at every time step and substitute it by a classical propagation of wave function parameters on the same footing with the nuclei.le3 This is not an actual propagation of the wave function, since no quantum mechanical propagator is involved (e.g., an initially real wave function stays real and does not de- velop imaginary phase factors), and therefore is in contrast to other methods4 which are set up within the framework of time-dependent Hartree-Fock theory. Rather, this treatment can be viewed as a continuous reoptimization of the wave function in order to keep it close to the true Bom- Oppenheimer solution.5-7

The resulting speed up made several applications and further method development possible;8-‘0 most of it, however, was confined to the domain of density functional theory (DFT) and plane wave basis sets. Some groups tried a semiempirical treatment,” elaborate improvements upon the tight-binding model, I2 or ab initio Hartree-Fock without nuclear dynamicsI Only very recently, an ab inifio

Hartree-Fock (HF) treatment with Gaussian basis sets was used by FieldI and Mohallem” for geometry optimization and by our groupI for AIMD.

With this paper, we report the implementation of an ab initio molecular dynamics scheme which has roots in the same general ideas as the methods referenced above, but goes beyond both DFT and simple HF theory by operating on the generalized valence bond (GVB) level. Therefore, we can treat open shell cases as exact eigenfunctions of spin (in contrast to DFT), include important parts of electron correlation, and describe the breaking of chemical bonds essentially correctly (in contrast to HF). The traditional Gaussian basis sets employed here should describe finite systems like clusters and molecules better than the plane wave bases with periodic boundary conditions used in many of the above studies.

We give examples for two characteristic applications of this method to a small metal cluster: simulation of free dynamics and search for the global minimum on the potential energy surface by coupling the AIMD to a simulated annealing scheme. Our version of AIMD makes it easy to follow the dynamics of different eigenstates of total spin, which leads in this case to a markedly spin-dependent dissociation dynamics that would not be observed in classical simulations. The different behavior of the spin states is easily rationalized pictorially by looking at the occupied electronic orbitals involved in each case.

The use of ab initio molecular dynamics simulated annealing (AIMDSA) for geometry optimization also opens up new possibilities over traditional methods. Traditional quantum mechanical optimizations can locate local minima; but for atomic clusters these methods break down because the number of local minima grows exponentially with the number of atoms.” Furthermore, local minima are quite meaningless without knowledge of the overall features of the surface. Simulated annealing18 has been in-

J. Chem. Phys. 97 (9), 1 November 1992 0021-9606/92/ 216569-l 0$006.00 @ 1992 American Institute of Physics 6569 Downloaded 15 Nov 2001 to 128.97.45.38. Redistribution subject to AIP license or copyright, see http://ojps.aip.org/jcpo/jcpcr.jsp

6570 B. Hartke and E. A. Carter: Valence bond molecular dynamics

traduced to remedy such situations. The essence of this method is the following: repeated sampling of the potential energy surface (using a random walk or a suitable deter- ministic algorithm, e.g., the classical equations of motion, as in this case) is subjected to a control parameter (the “temperature”). This parameter is varied such that initially the sampling covers the whole surface (which makes the system forget its initial conditions), but is then nar- rowed down slowly to lower energy regions (in the above cases, by reducing the probability for steps to higher energies or by lowering the kinetic energy of the system). This general optimization procedure can be guaranteed” to find the global minimum with moderate conditions on the cooling schedule and an infinite (in practice, very large) number of propagation steps. The combination of simulated annealing with AIMD fully exploits the advantages of both methods, the independence of any a priori knowledge of the surface and the independence of the initial conditions. In principle, therefore, one single run of AIMDSA should suffice to find the global minimum. In practice, however, the propagation is very costly and has to be made as short as possible. This deprives the method of the guarantee to find the global minimum, but it has been shown” that such a procedure is still more successful and effective than a series of traditional optimization runs. Accordingly, it has been used in many applications of DFT-AIMD.8’10

In Sec. II, we describe all the pertaining details of our theoretical approach. Sections III and IV present results for free dynamics and AIMDSA geometry optimization, respectively. Conclusions are given in Sec. V.

II. COMPUTATIONAL METHOD

A. Equations of motion

We expand the molecular orbitals bi in a basis of Gaussian functions X~ centered on the nuclei, $i=Xpc,x . In the GVB perfect pairing (GVB-PP) approximation, $0

each nonorthogonal GVB pair orbital &,+ #ii can be written as a linear combination of orthogonal natural orbitals #lp$*jaS$;,*j= (Uti/2#ljf Uiy+*j)/(UV+ ~~j)1’2.Thusfor a fixed nuclear geometry {RI) and a fixed Gaussian basis set, the shape of the wave function can be fully described by the HF-SCF coefficients {Cpi} and the GVB coefficient ratios {aj}, where aj=Uzi/olj (it suffices to consider the a’s, since the u’s are subject to the additional normaliza- tion constraint dj+dj= 1). In order to make the wave function follow the dynamics of the nuclei, the wave function parameter sets {c,i} and {ai) are combined with the nuclear coordinates {RI} into one system undergoing joint classical dynamics.5-7 Since the SCF and GVB coefficients are per se massless, we have to give them fictitious masses mSCF and mGVBI respectively, in order to subject them to classical dynamics. This establishes the classical Lagrang- ian in Eq. ( 1)

L= F iM,R:+ C f m.yc&+ $’ f rnGv&i Pi

--E(CRII,CcpJ,{ajl) (1)

with nuclear masses Mp This Lagrangian leads in the usual way to the equations of motion (2)

. . MrRI= -VIE,

msc+$i= - 2 C GPpyf (2) Y

mov&j=-2[2CZj(Y$,--Y/;1)+(l-cY~)K{z]/(l+C$)*.

As forces on the nuclei, we use either the full forces -VIE or Hellmann-Feynman forces, Eq. (3))

aE H&Fey

a4

a vNN

+ dRI t (3)

where VNN is the internuclear repulsion potential. The forces on the SCF coefficients in Eq. (2) involve the Fock matrix element pp,, for shell n of orbital i in the atomic basis x,, xy. For simplicity, we assume here that pp,, al- ready includes the proper dependence on GVB coefficients and shell occupation numbers.20 The forces on the GVB coefficients in Eq. (2) are expressed in terms of Y$ and Kj2 matrix elements over molecular orbitals for orbitals i= 1,2 in GVB pairj. They are defined as*’

Closed

Yii=hij,tj+ C (wq,ij-Kq,ij) +$ Jg,g Q

pair Ope”

+ kzj &W~p,ij-Kkp,ij) + c (Jr,i+%& I (4)

and

K<*=Klj,*j 2 (5)

where h contains the one-electron terms, and J and K are the usual Coulomb and exchange integrals. The full forces (or the Hellmann-Feynman forces) on the nuclei as well as the matrices pU,,, Y~i, and Kj2 can be obtained directly from marginally modified programs.*‘.**

6. Orbital orthonormality

standard electronic structure

The initial condition of orbital orthonormality, Eq. (6)) of course, must be preserved throughout the propagation

(4il #ji> = C c/IFV~pY=blj (6) PV

(where Spy is the overlap integral between the basis functions x, and xv). This puts constraints on the SCF coefficients; we ensure their fulfillment after each time step by

J. Chem. Phys., Vol. 97, No. 9, 1 November 1992

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B. Hartke and E. A. Carter: Valence bond molecular dynamics 6571

applying the standard SHAKE procedure23 in the following manner (see also Ref. 6).

The constraints can be written in the following way:

u’j= $ qLFvpp--sLy=o* (7)

They produce an additional term --BiJljp, in the La- grangian (where the A;s are Lagrange multipliers), and therefore show up in the equations of motion for the SCF coefficients (2) as an additional term Zfisc= - xjnii(&ri/ a~,,), which can be interpreted as a sum of “constraint forces.” Now, we assume that the constraints are fulfilled at time t and propagate the dynamic system for one time step At without considering the constraint force terms. The new constraint expressions will then in general be different from zero

Gu(t+At)= CC,i(t+At)~~~(t+At)S,,(t+At)--i~O P” (8)

(here the tilde denotes values obtained by unconstrained propagation). However, we know that the constraint expressions after application of the constraint forces should equal zero, and we can express them in a Taylor series through first order,

I

In the case i#j, there is a second equation analogous to Eq. ( 11) but with the substitutions i-j and ~+PY. Equa- tion ( 11) would yield the correct constrained coefficients directly if there were only one constraint on each orbital and if Eq. (9) were exact. Instead, Eq. (9) is correct only to first order, and several constraint equations apply to the coefficients of each orbital as soon as more than one orbital is taken into account. Therefore, the equations have to be iterated to convergence; in our examples, typically 5-10 iterations are sufficient to converge all constraints to within a tolerance of 10v6. In practical applications, we can save memory for storing the overlap matrices SPY by the approximation S,,,( t+ At) &,,(t), which is valid as long as the time step At is short compared to the movement of the nuclei.

C. Numerical propagation

The simple Verlet algorithm,24+25 Eq. ( 13), is used to propagate q= RI, cpi, oj (with the appropriate masses m,) according to the above forces

(AQ2 aE q(t+At)=2q(t)-q(t-At)---

mq a4 [’ (13)

We chose the easily programmable simple Verlet scheme over velocity Verlet,25’26 because we have no need to know

a,(t+At)=O

=iQt+At> + c ao,(t+At)

p a&i(t-tAt) [CJt+At

aG&+At) -$i(t+At)lj+ G ac ,tt+ht)

VJ

x [c&+At) -Fvj(t+At)]i. (9) The change [c,i(t+Ar)-~~i(t+At)]j of the SCF coefficient cPi due to the constraint aij is brought about by the constraint force gPii and can therefore be written as

(At)* aoG(t) ( At)2 -- == gdf) = -% acpi( t) mSCF * (10)

Equations (S), (9), and ( 10) allow for the evaluation of the $‘s, which in turn yields the constrained values for the SCF coefficients at time t + At,

[c~i(t+At)]j=[~~i(t+At)]j-I2-bCcVj(t)S,,(r), (11) v

where we have introduced a modified ,lb according to

,,=PE a ..= ZkFki(t+At)Clj(t+At)Skl(t+At)--~ ‘J mScF ” x/p,[Clj(t+At)Cpj(t) +~i(t+At)Cpi(t)]Snl(t+At)S,p(t) *

(12)

I

exact velocities at any point. When velocities are needed for “cooling” during an SA run (see below), it is sufficient to approximate them by simple two-point finite differences.

The time step At was optimized to a value of 5.0 atu ( 1 atu=2.418 884 28X lo-l7 s) or approximately 0.12 fs. This leads to conservation of total energy (including the fictitious kinetic energies of the wave function parameters) on the order of 5 X 10e6 hartree over a period of 1000 time steps (0.12 ps). Conservation of total linear and angular momenta in the nuclear dynamics was not checked numer- ically, but is at least fulfilled to graphical resolution of nuclear geometry plots over any calculated time period. Initial conditions were always chosen with total linear and angular momenta equal to zero.

The price one has to pay for avoiding the rediagonal- ization of the wave function at every time step by applying the above equations of motion (2) is a slow drift of the wave function away from the true Born-Oppenheimer solution at each nuclear geometry.2’5P6 This makes it necessary to rediagonalize the wave function after longer intervals of propagation; this also takes care of the adiabaticity problem.7P9*27 We found it sufficient to do these rediagonal- izations every 500 steps in HF cases and every 100 steps in GVB cases. Of course, this is a static scheme which includes a wide safety margin, and a dynamically adaptive scheme would be an improvement2*

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D. Fictitious masses

Possible values for the fictitious masses of the SCF and GVB coefficients are limited from above by the require- ment that we want to reproduce true nuclear dynamics, i.e., we want the fictitious kinetic energies of the wave function parameters to be negligible compared to the potential and kinetic energies of the nuclei (see also Refs. 5-71,

tern, its location on the potential energy surface and its temperature. For higher efficiency, a dynamical updating scheme or, ultimately, a dynamical multiple time step scheme would be desirable.28

F. Simulated annealing: General considerations

&t4La1~ (14)

where

1 41,~ = c 5 mscti$+ 1 ’ mGv&,

Fti i 2

l&d= F i ~J~:+E(CRI},Cc,i},Caj)). (15)

This also ensures slow transfer of kinetic energy between the subsystem of the nuclei and the subsystem of wave function parameters. A lower limit for the fictitious masses is given by the practical consideration that we want the time step to be as large as possible but still small enough to follow the high frequency oscillations of the wave function parameters’-7 fairly accurately. In several short test runs, the nuclear dynamics turned out to be independent of changes of the fictitious masses by 2 orders of magnitude; in all simulations reported here we have used a value of 300 a.u. or approximately 0.16 amu as the fictitious mass for each GVB and SCF coefficient.

The above fully characterizes our AIMD methodology with respect to applications in the free dynamics regime. Coupling of AIMD with SA for geometry optimization is most simply achieved by a sequence of free propagations linked by “cooling steps”‘8~19 (i.e., scaling of the instantaneous nuclear velocities by a factor in the region 0.5-0.95). As mentioned in Sec. I, one would like to have long periods of free dynamics (for a full sampling of the accessible phase space regions) and slow cooling (in order to fulfill the guarantee of reaching the global minimum), but this is prevented by the expense of AIMD. Although some work has been done to improve upon traditional SA,29 we retain our simple approach here, since these calculations are intended as a general demonstration of feasibility of AIMDSA in the GVB picture. The example given in Sec. IV was generated by periods of free dynamics over 500 steps and a cooling factor of 0.90.

E. Propagation approximations

Since the nuclei are typically 2 orders of magnitude heavier, we are dealing with a dynamical system containing a huge mass disparity. Therefore, it is advantageous to use a multiple time step propagation scheme, in order to save time on recalculation of the forces on the nuclei and the integrals over basis functions (which change with each movement of the nuclei). The implementation of such a scheme will be reported elsewhere.28 Here we have exploited the large mass disparity only in an approximate, crude fashion: the time step is the same for both the nuclei subsystem and the wave function parameter subsystem, and the movement of the nuclei is slow on this timescale (see above and Refs. 5-7). Thus also the forces on the nuclei and the integrals over basis functions are changing slowly and need not be recomputed every time step. From short test runs, we learned that there is no appreciable loss in accuracy of the nuclei dynamics when the integrals are updated every fifth step and the forces every tenth step, where we use a cheap linear extrapolation of the forces between recalculations. The skipping of integral recalculation introduces artificial spikes in the fictitious kinetic energy of the SCF coefficients and corresponding steps in the electronic and potential energies, but these features are mi- nor departures from a baseline that remains correct (see Sec. III). This scheme includes a wide safety margin, since it is static, and its validity obviously depends on the sys-

Two other characteristics of SA allow for further optimization of this procedure3’ and should be mentioned here: initially, the system has to have a high temperature, so as to eliminate the dependency on the initial conditions. In this stage, the accuracy of the forces on the nuclei is not critical, since the dynamics are governed mainly by the gross, qualitative features of the potential energy surface. These are in general modeled acceptably by Hellmann- Feynman forces without “Pulay corrections”.‘0*16~3’ There- fore, the initial stages of an SA mn can be accelerated by using Hellmann-Feynman forces; one should switch to full forces as soon as the error in the Hellmann-Feynman forces becomes comparable to their magnitude. The inac- curacy of the Hellmann-Feynman forces destroys the conservation of total energy and total linear and angular momenta, but this poses no severe problems: the convergence of the finite differencing scheme used can be tested by total energy conservation with full forces, and the time step found can be applied to propagations with Hellmann- Feynman forces,16 since their general magnitude is not ba- sically different from full forces. The nonconstant total angular momentum can lead to a drain of kinetic energy into an “orbiting motion,” which disrupts the regular cooling scheme and may lead to misinterpretations of the temporal behavior of the kinetic energy (the orbiting motion does not affect internuclear distances and therefore the temperature of the cluster does not reflect the area of accessible phase space anymore). But in the AIMDSA context, molecular dynamics is merely an efficient means of sampling the potential energy surface; hence we can avoid the orbiting motion by correcting the nuclear velocities back to zero total angular momentum at each step.30

In the last stages of SA, the kinetic energy is very low and approach to the minimum is s~ow.~***‘~ Obviously, one can shortcut the fmal optimization stage by switching from



SA to a traditional optimization scheme like the conjugate gradient method,32 as soon as the system is unable to escape from the basin of attraction of the minimum. In the exemplary application of AIMDSA given in Sec. IV, we have omitted these finer points by using full forces throughout the propagation and by bringing the system from a random 3D geometry to the proximity of the global minimum in two dimensions.

G. Improved simulated annealing for metal clusters

Our specific example of singlet Na, poses even more problems, which can be expected to occur frequently in main group and transition metal clusters. On the GVB level, the potential energy surface is very flat and has wide open channels to dissociation. The two-dimensional, rhombohedral geometry, which is generally assumed to be the global minimum,33’34 can be traditionally optimized to the half-axis lengths of 3.217 and 1.761 A, respectively, and the energy of -0.746 254 hartree [using a GVB (2/4) wave function2’ and the basis set and ab initio coreless Hartree-Fock (CHF) effective core potential from Ref. 35, explicitly describing only the four valence electrons]. The optimized energy of two infinitely separated singlet Na2 is on the GVB( l/2) level -0.744459 hartree, or only 0.001 795 hartree above the rhombus. This energy difference corresponds to a temperature of only 94 K. SA, however, involves starting at a high temperature; therefore we have to rigorously avoid dissociation of the cluster even at temperatures considerably higher than 94 K. This can be achieved by confining the cluster in a spherical, reflecting boundary around the center of mass, which is small enough to keep the atoms in the interaction region but large enough not to influence possible final geometries. The spherical, reflecting boundary has the additional virtue of leaving the conservation of total angular momentum in- tact;25 like the more commonly used cubical boundary, however, it destroys conservation of total linear momentum. We have chosen a spherical, reflecting boundary with a radius of 4 A in our example in Sec. IV.

Preventing dissociation, however, is not enough to ensure viability of SA for GVB(2/4) singlet Na,. Studies have shown that there is a critical temperature29 in the SA process which has to be passed slowly in the cooling schedule, and which should be linked to the energy differences characterizing the prominent features of the surface. In this case, there is an energy difference corresponding to only 94 K between vastly different geometries, and preliminary tests indicate that there may be no significant barrier in between. This means, that a long period of annealing has to be spent around a temperature of 94 K (or possibly slightly higher). But at such a temperature, the nuclei are moving very slowly and covering significant parts of the accessible potential energy surface becomes prohibitively costly. This forces us to introduce another modification: we would like to scale the potential energy surface such that the energy differences become much larger, which in- creases the critical temperature and therefore the velocity of the nuclei at the critical stage of SA. Obviously, such a

scaling is equivalent to a constant scaling of the forces on the nuclei, which in turn is equivalent to a constant scaling either of the time step and the fictitious masses or of the nuclear masses. For example, at 100 K and with the true mass of 22.9898 amu= 409.29 a.u. for each Na atom, each degree of freedom can move only 2.4~ low4 A per step on ave!age and would need approximately 4000 steps to cover 1 A. Scaling the masses down to 300.0 a.u. makes it possible to cover 1 A in only 340 steps at the same temperature. We use this scaling in the example given in Sec. IV. Since the fictitious masses still retain the same value of 300.0 a.u., this means pushing the scaling idea to the limit, and we expect fairly rapid exchange of kinetic energy between the true and fictitious degrees of freedom. But on the other hand, we would like to emphasize again that in MDSA we are not interested in reproducing a phys- ically real trajectory of the nuclei. The equivalent idea of abandoning real dynamics in AIMDSA by making the fictitious masses larger to allow for a longer time step was hinted at in the literature’ but apparently never realized.

For clusters or molecules with deeper potential wells at the GVB level of theory, the mass scalings and reflective boundaries may not be necessary, as there will exist a large enough temperature range through which to sample prop- erly the potential energy surface.

III. TEST RESULTS FOR FREE DYNAMICS

In order to demonstrate the feasibility of GVB-AIMD, we have simulated the free dynamics of singlet and triplet Na4 on the GVB( 2/4) and GVB( l/2) level, respectively, using the same basis set and effective core potential as mentioned in Sec. II G [GVB(m/n) denotes a GVB wave function containing m GVB pairs which are described by n natural orbitals]. Both propagations start from the same, arbitrarily chosen initial geometry: a rectangle that was slightly distorted from planarity to avoid artifacts arising from symmetry conservation (the exact geometric parameters are irrelevant since we want to demonstrate qualitative effects only). Initial nuclear velocities are set to zero in both cases. The initial total energy is well above the dissociation limit mentioned in the previous section. The dissociation dynamics, however, are quite different for the singlet and the triplet (cf. Fig. 1): the dissociation products are two ‘Na2 and ‘Na2 plus two Na atoms, respectively. This can be rationalized easily from the individual wave functions (Fig. 2): each of the depicted orbitals accomo- dates one electron. In the singlet, the orbitals belonging to the same pair overlap, while orbitals of different pairs are orthogonal to one another and hence have a node in between them. As a general principle, the electronic kinetic energy can be lowered by decreasing the slope of the wave function at a node. This generates a force which drags the pairs of Na atoms associated with the GVB pairs away from the nodal plane-and this is exactly the behavior of the nuclei in the singlet. The difference in the triplet lies in the open shell orbitals: they correspond to a bonding and an antibonding orbital for one pair of sodium atoms, and both are singly occupied. The open shell orbital of lower



6574 B. Hartke and E. A. Carter: Valence bond molecular dynamics ain.let NE4 dynamics I,..: ib: ~=~,~: ic: ~=,~,*: 0 0 0 .

0 0

0 0 l t = 0,‘ t=wJ4 0 l d e f t = lso,a 0

triplet rh.4 dynamics

FIG. 1. Temporal behavior of the nuclear geometry in GVB singlet and triplet Na, propagation: Panels (a), (b), (c) show singlet Na, developing from a very slightly distorted rectangle of four atoms to two distinct Na, pairs. In panels (d), (e), (f), triplet Na, evolves from the same starting geometry towards Na,+2 Na.

energy has essentially the same structure as the GVB pair orbitals. The one with higher energy, however, has an additional node between the sodium atoms. By the same ar- gument as above, this feature leads to a force pushing this sodium atom pair apart.

Figure 3 shows the temporal behavior of some characteristic quantities in the propagation of singlet Na, (they behave qualitatively similarly in the propagation of the

triplet). In accord with the virtually barrierless dissociation and the high initial potential energy, the total kinetic energy of the nuclei [Fig. 3 (a)] rises rapidly and monoton- ically. The total energy [Fig. 3(b)], including the real kinetic and potential energy of the nuclei and the fictitious kinetic energy of the wave function parameters, remains constant within graphic resolution throughout the propagation, reconfirming the appropriate choice of the time step. The fictitious total kinetic energy of the SCF coefficients [Fig. 3 (c)] remains three orders of magnitude below the real total kinetic energy of the nuclei [Fig. 3(a)]. As mentioned in Sec. II, the spikes are an artifact of the integral recalculation skipping scheme which has no bearing upon the nuclear dynamics (a very costly calculation without this skipping scheme would exactly reproduce the baseline of these spikes). The fictitious total kinetic energy of the GVB coefficients [Fig. 3(d)] is 2 orders of magnitude smaller than the fictitious total kinetic energy of the SCF coefficients. Therefore, Eq. ( 14) is fulfilled, and we can be confident that the nuclear dynamics represent real trajec- tories.

IV. TEST RESULTS FOR SIMULATED ANNEALING

As a test of GVB-MDSA using reflecting spherical boundary conditions and scaling of nuclear masses (cf. Sec. II), we apply this scheme to singlet Na,. Other test

singlet wavefunction

r

GVB pair 1

a L \

1

GVB pair 2

- / / i

_.., :,;{ @ ,:’ : ‘.__.’ @

c 0 :

C \ \

I

‘1 i 0 \

,..... ;,i: 0

:\ @ ‘.._.’ I @ \

d \. \ -

triplet wavefunction

GVB pair

,--- --\ \

e

1 open shell orbitals

FIG. 2. Contour plots of occupied orbitals in the initial wave function of singlet and triplet Na, for the geometry shown in Figs. 1 (a) and 1 (d) at t=O ps: Panels (a), (b) and (c), (d) show GVB pairs 1 and 2, respectively, of singlet Na, at the initial nuclear geometry. Panels (e), (f) depict the GVB pair of triplet Na, and panels (g), (h) its open shell orbitals. The four nuclei are at the loci of highest density in each panel. Solid lines and short-dashed lines denote positive and negative amplitudes; long-dashed lines show the location of nodal planes. Contours range from -0.05 to +0.05 a.u. at intervals of 0.005 a.u. The curious structure of the nodal plane in the left part of panel (g) is due to difficulties of the plotting routine with low wave function values.




Ekin,SCF

/I@ hartree

7-- -i---r W-1-I ;b

- 0.73G i

I %fal I /hartrre

: - 0.738 F

1

i- i

1 , / i -i-- L 1-d 100 tlfs 200

d

Ekin,GVB II II /IO+ hartree

FIG. 3. Temporal behavior of different parts of the total energy during propagation of singlet Na4: Panel (a) shows the kinetic energy of the nuclei, panel (b) the total energy, panels (c) and (d) the fictitious kinetic energies of the SCF and GVB coefficient subsystems, respectively. Note the different scales of panels (a), (c), and (d)! The prominent spikes in panel (c) are entirely an artifact of the skipping of integral recalculation, see the text; the proper curve is given by the baseline.

runs indicated that the potential energy surface is quite flat in the two dimensions of the global minimum structure (the planar rhombus), but presumably steeper in the third dimension perpendicular to these two. Therefore, one easy goal to achieve in a MDSA run is bringing singlet Na, from a 3D geometry to a 2D geometry. We have chosen an arbitrarily distorted tetrahedral geometry with a kinetic energy of the nuclei of 0.002 hartree (random velocities corresponding to an instantaneous temperature of approximately 105 K) as initial conditions and subjected the system to the cooling procedure given in Sec. II. The resulting behavior of the nuclear geometry can be seen in Fig. 4. Clearly, the system ends up in a planar geometry (with some residual vibration in the third dimension), which is, however, quite different from a rhombus: three atoms are arranged in an acute triangle with the fourth one sitting near the center of the triangle. Due to the flatness of the potential energy surface in these two dimensions, the re-

sidual forces are not large enough to be used in a traditional geometry optimization. To make the system end up in the (presumably) correct final geometry of the rhombus, two different strategies can be envisioned: (a) the whole trajectory could be repeated one or more times with different initial conditions and a refined cooling schedule; or (b) the final geometry from this run could be used as a starting geometry for “reannealing,” i.e., a continuation run starting with rerandomized nuclear velocities at a level of kinetic energy higher than that at the end of this run.

In this AIMDSA trajectory, the real total kinetic energy of the nuclei is only a factor of approximately 5 larger than the fictitious total kinetic energy of the SCF coefficients (not shown; the fictitious total kinetic energy of the GVB coefficients is again 2 orders of magnitude lower than that of the SCF coefficients, as in Sec. III). This classifies the nuclear dynamics as on the borderline between true dynamics and completely fictitious geometry sampling (as


6576 6. Hartke and E. A. Carter: Valence bond molecular dynamics

singlet Nq simulated annealing I I’ I * I

C -0.73 k 11 Epot / hartree

1 -0.743

0 l 0 0 9

; a a

t = Ofs t =42fs t=84fs

d e f

i

t . .

i

00

3

1 1 ,I! :’ /VI, ? [ - -0.744

-0.74 t ;

,I;

,I \/\/I(

111; 6 @“A7 i I x -I- 0.745 C’ / \r\ I I li” V-L I

t = 127fs t = I69.b t = 212fs

) b ,A, /

0 100 200 300 400 500

t1f.s

h j

l * 0 FIG. 5. Potential energy during simulated annealing of singlet Nq: This plot shows the temporal behavior of the potential energy (electronic energy plus internuclear repulsion) as seen by the nuclei at each instant of the propagation. The right part of the plot is enlarged by a factor of approximately 6 for clarity.

t = 254fs t = 29Gfs t = 508fs

FIG. 4. Simulated annealing of singlet Na,: The distorted tetrahedral starting geometry is flattened out to a planar geometry in panels (a)-(h), which corresponds to snapshots of the nuclear dynamics taken at intervals of approximately 42 fs, and panel (j), which shows the nuclear geometry at the end of the run. In the pseudo-3D rendering of the cluster, larger circles denote atoms close to the viewer, while those farther away appear smaller.

discussed in Sec. II, this is acceptable for SA). Due to the similar masses for the dynamic subsystems of the nuclei and the wave function parameters, there is a large coupling between them, inducing rapid transfer of kinetic energy (recall that this is not found when the masses of the nuclei are 2 orders of magnitude larger than those of the wave function parameters, as in Sec. III). Since the periodic reconvergence of the wave function resets the velocities of the wave function parameters (independent of their previous values), this energy transfer amounts to a substantial drain of kinetic energy away from the nuclei. This occurs mainly during the first 100 fs of the propagation (the “induction period”); it disturbs the intended cooling schedule and almost turns it into a quench. In later stages, the fictitious kinetic energy becomes smaller compared to the real kinetic energy of the nuclei, reducing this energy transfer considerably. Since the geometry of the cluster is still three-dimensional in these later stages, the induction period presumably has no decisive influence on the outcome of this particular SA trajectory. If the induction period turned out to fully disrupt the cooling schedule, it could be avoided either by retaining the fictitious kinetic energy upon wave function reconvergence, or by reintroducing the kinetic energy lost in the wave function parameter subsystem back into the subsystem of the nuclei.

The temporal behavior of the potential energy (cf. Fig. 5) exhibits interesting features: the fairly large, unsystem- atic peaks are an indication of the large coupling between nuclei and wave function parameters. The induction period characterized above can be seen to end when the system settles down at a potential energy value of approximately -0.744 hartree. At this point, the nuclear geometry is still clearly three dimensional. In the ensuing period up to approximately 250 fs, the nuclei swing through a planar structure and then tend back to a different planar arrange- ment [Figs. 4(d)-4(g)]. This is reflected by a two-step decrease of potential energy (Fig. 5). At times later than 300 fs, the nuclear geometry remains essentially planar and unchanged (apart from a vibration of the central sodium atom through the plane of the other three), while the potential energy exhibits a slowly damped oscillation about its lowest value, in other words, the system cannot escape from the basin of attraction of this (local) minimum (the slowness of the damping makes it essentially impossible to refine this structure by continuation of the SA run, cf. Sec. II). We believe that these signatures in the potential energy curve are fairly generalizable and will provide valuable aid in optimizing larger clusters.

Preliminary tests with reannealing of the final geometry reached in this trajectory indicate that random nuclear velocities corresponding to an instantaneous temperature of 100 K suffice to liberate the system from the local minimum and make it move towards the rhombohedral structure without significant departure from planarity. Tradi- tional geometry optimization from several points on this continuation trajectory, however, still proves to be difficult due to the flatness of the potential energy surface.



V. CONCLUSIONS

In this paper, we have presented the first implementation of AIMD on the GVB level. Details of the method have been given in Sec. II. In Sets. III and IV we have provided first application examples of our program, both to free dynamics and to geometry optimization by help of SA.

The example for AIMDSA given in Sec. IV indicates that our parameters are not yet optimized; but in spite of its crudeness, the algorithm was able to guide the system to the neighborhood of the global minimum. Thus it fits the purpose of this paper as a proof of principle and a first step in the right direction. Parameter refinement will be done in subsequent work.

Naturally, our method is computationally very de- manding. It would be premature to discuss computer timings at this stage of development, but one aspect of com- parison to applications’ of AIMDSA to clusters in the DFT picture with plane wave basis sets is worth mention- ing: the plane wave basis is independent of the positions of the nuclei; for this reason there is no need to recalculate basis function integrals as the nuclei move during propagation. On the other hand, a very large number of plane waves is needed to describe finite systems appropriately. We use Gaussian basis functions centered on the nuclei; therefore much fewer basis functions are sufficient but recalculation of basis function integrals becomes necessary. It turns out that these tendencies cancel out approximately, and overall timings have the same order of magnitude.

The application examples given in this paper are intended as proofs of feasibility of the calculations, and they help us gain experience and valuable insights for applications to interesting, larger systems. Our method brings three well-established areas of research together (GVB, MD, and SA) and thereby creates a new area: the use of ab initio molecular dynamics with a molecular quantum mechanical wave function to do simulated annealing as a means of an unbiased geometry optimization of atomic clusters.

ACKNOWLEDGMENTS

This work was supported by the Office of Naval Re- search (Grant No. NOO014-89-J-1492). E. A. C. also ac- knowledges support from the National Science Foundation and the Camille and Henry Dreyfus Foundation through their Presidential Young Investigator and Distinguished New Faculty Award programs. B. H. is much obliged to the Fonds der Chemischen Industrie for a Liebig scholar- ship. We also thank D. A. Gibson for help with program testing and graphics visualization.

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