electrostatic solvent effects on the electronic structure of ground and excited states of molecules:...

Electrostatic Solvent Effects on the Electronic Structure of Ground and Excited States of Molecules: Applications of a Cavity Model Based upon a Finite Element Method

Thomas FOX,' Notker Rosch,'* and Randy J. Zauhas 'Lehrstuhl fur lkzoretische Chemie, Technische Universitat Munchen, W-8046 Garching, Germany, and Biotechnology Institute, Wartik Laboratory, Pennsylvania State University, University Park, Pennsylvania 16802

Received 24 July 1992; accepted 23 September 1992

We present investigations on the use of dielectric continuum models for the self-consistent description of electrostatic solvent effects on the ground state of a molecule and on excitation energies. The electronic structure calculations have been carried out in the framework of the INDO and INDO/S-CI method, respectively. We compare the performance of three implementations of the cavity model that all allow an arbitrary shape of the solute cavity. The procedures differ in the effort spent on the description of the charge density at the cavity surface. Two procedures in the vein of MiertuS, Scrocco, and Tomasi (MST) rely upon point charges to model the reaction field and differ in the way the cavity surface is modeled. While one implementation divides the surface into flat triangular patches, the improved version uses curved triangles. Alter- natively, we investigate a finite element method (FEM) for the calculation of the surface charge density. Applications to rod-shaped organic molecules (including their charge transfer excitations) illustrate the superiority of the improved MST formalism over the primitive one, as it exhibits faster convergence of the results with increasing node density on the cavity surface. The FEM approach, which also employs curved surface patches, leads to afurther improvement as it needs less computational effort, especially in the treatment of excited states. 0 1993 by John Wiley & Sons, Inc.

INTRODUCTION

Recently, substantial progress has been achieved in the formulation and application of procedures de- scribing the interaction between a solvated molecule and its surrounding solvent.' Typically, a quantum mechanical description of the electronic structure of the solute is combined with a model representation of the solvent. The total interaction between solute and solvent can be separated into several contributions, the electrostatic part of which is the most thoroughly investigated?-" However, there have also been first attempts to include other contributions, such as the dispersion (or cavitation) interaction,18-20 as well as specific, directed interactions between solute and solvent molecules?122 It is the aim of the present work to discuss an accurate and efficient self-consistent procedure for evaluating the electrostatic solvent effects within a quantum mechanical description of the electronic structure of the solute.

Electrostatic continuum methods consider the solvent as a polarizable isotropic dielectric environ-

'Author to whom all correspondence should be addressed.

ment in which a cavity is prepared that contains the solvated molecule of interest. The solute charge distribution polarizes the dielectric and gives rise to a reaction field acting back on the solute. The available variants of this approach have been classified prof- itably according to several aspects.' One criterion discriminates between treatments where the charge distribution of the solute is kept and those where the reaction field is allowed to modify the solute Hamiltonian self-consistently~~6~8~g~11~1z~17~z2 In the latter formulations, one achieves an improved description of the reaction field at the expense of dealing with a new charge distribution in each solute-solvent self-consistency cycle whereas in the former one has to solve the classical electrostatic problem only once.

Another obvious classification criterion is derived from the shape of the cavity employed. The most simple choice is the use of a constant coordinate surface, in practice, a sphere or an e l l i p s ~ i d ? , ~ , ~ ~ ~ For this special case, an analytical solution of the electrostatic problem is available.23 However, for arbi- trarily shaped molecules, especially for large organic compounds, such a shape restriction seems often rather unphysical. This becomes evident in the problems associated with choosing the size parameter(s) of the cavity.821Z2Z6 A physically more appealing al-

Journal of Computational Chemistxy, Vol. 14, No. 3,253-262 (1993) 0 1993 by John Wiley & Sons, Inc. CCC 0192-8651/93/030253-10

254 FOX, ROSCH, AND ZAUHAR

ternative is to use a realistic cavity that is adapted to the shape of the solute, e.g., a set of interlocking spheres surrounding the atomic ~ e n t e r s . 6 ~ ~ ~ " ~ ' ~ ~ ' ~ Be- cause in this case no analytical method is feasible for generating the reaction field, one has to resort to a numerical procedure for the solution of the en- suing electrostatic problem. The solvent effect is most conveniently incorporated into the solute Ham- iltonian via a self-consistently determined "external" charge density v(r) located at the cavity surface. Many implementations of this approach, which was f i s t presented by MiertuS, Scrocco, and Tomasi (MST): employ a distribution of point charges to model the surface polarization charge density.6,9J6J722 However, past MST implementations were plagued by the requirement of a large number of point charges, even for molecules of moderate size, to arrive at reliable results for the solvent effect. This disadvantage entails a significant computational effort that becomes especially noticeable within a semiempirical electronic structure method, where it tends to dominate. Here, we present a re- finement of our previous MST implementation that exhibits a drastically improved convergence with increasing node density. An alternative scheme utilized a finite element method (FEM) that relies upon a more sophisticated representation of the surface charge density. It promises a significant improvement over the crude, but straightforward implementations based upon a discrete representation of the polarization charge density. Such a method has been developed by one of us15 and applied to the evaluation of solvation energies of large biomolecules in molecular modeling investigations. This FEM procedure will be employed here for the first time self- consistently in conjunction with electronic structure calculations.

In this article, we continue our series of investigations on the use of a self-consistent electrostatic cavity model within the framework of the INDO/ S-CI rneth0d.'~2~2~ Here, we compare the simple point charge approach to the finite element method. We do this by calculating the solvent effect on ground-state properties and on the electronic excitation spectrum of some organic molecules. We be- gin with a series of rod-shaped donor-acceptor compounds where the two functional units are separated by rigid msystems of varying extension. Using a large electron-transfer compound as an example, we then show that the FE method is, in principle, able to cope with large solute molecules. We focus on performance differences of the two methods, but leave aside a thorough physical interpretation of the results as a pure electrostatic description of the solvent effect (underlying the cavity model used here) has been shown to have severe inherent deficiencies if a quantitative description of solvatochromic shifts is desired.l7z2

METHODOLOGICAL CONSIDERATIONS

To arrive at a realistic cavity shape, we decided to base it upon Connolly's notion of the "solvent ac- cessible surface."2s Further, a representation of the surface polarization charge density has to be selected. As mentioned above, the two methods to be compared in this study differ mainly in this way. Although both procedures have been described in detail e l ~ e w h e r e , 6 1 ~ ~ ~ ~ ~ ~ ~ it is appropriate to briefly re- view their main features.

In our variant of the straightforward point charge procedure, the dielectric interface of interlocking atomic spheres is approximated by N, triangular patches determined by N, nodes on the cavity surface. The underlying triangulation procedure has been described previo~sly?~ The charge density on each triangular patch is assumed to be constant and is represented by a single point charge of appropriate size, located at the midpoint of the patch. The start- ing ckarge distribution is derived from the electric field E [ p] caused by the molecular charge distribution p. The mutual polarization of the molecular and the surface charge density is taken into account via an iterative process where, in the mth cycle, one obtains the point charge of the jth element:

qy = g(€o)zj. (&p] + E[o"-']) A s j (1)

Here, Z[v"-'] is the field due to the surface point charges resulting from the (m - 1)th iteration, Gj de- notes the outer surface normal and Asj the area of the jth surface patch, and g(Eo) the well-known reaction field factor. The solute Hamiltonian is then modified by the presence of the "external" electrostatic potential

In the FEM procedure, the cavity surface is described by a set of curvilinear triangular patches that form a continuous, piecewise analytic surface. The same triangulation procedure29 is employed as above, but a total of 10 integration points is chosen for each triangular element. In addition to the three corner nodes, two points on each of the curvilinear edges and one point in the interior of the patch are selected. Based upon these 10 points, one defines a cubic interpolation for each surface element using Lagrange polynomials. Once the surface has been decomposed, an N, x N, matrix of interaction coef- ficients Kij is found by numerical integrati~n.'~ This coefficient matrix K effectively summarizes the electrostatic interaction of an arbitrary surface charge density with itself and depends only upon the set of N, nodes, i.e., upon the shape of the cavity and the triangulation of its surface. The detailed formulation of the K matrix is straightforward but somewhat lengthy. We therefore refrain from repeating the der-

ELECTROSTATIC SOLVENT EFFECTS 255

ivation of ref. 15. Given a (molecular) charge distribution d r ) in the interior of the cavity, the normal components of the corresponding electric field E [ p] at each _node are calculated+and stored in a column vector F [ p ) = (Fi[p] = (EJp] * ZJ. The column vector of nodal polarization charge densities is then found as the solution of the system of h e a r equations:

{I - fK}; = f&] (3)

where f = (1 - E ) / ( ~ T ( E + 1)) is a constant that depends upon the dielectric constant E of the solvent.15 Such an FE method is in particular advantageous in combination with an electronic self-consistent procedure where the molecular charge density p has to be determined repeatedly, e.g., when several excited states are to be treated self-consistently. All computational effort connected with the integration over the curved surface is performed only once and its result is stored in a “compact” form in the coefficient matrix K. In each iteration, only eq. (3) has to be solved; in our implementation, this is done by a GaussSeidel process.

The polarization charge density at an arbitrary point r on the molecular surface is then found by linear interpolation:

a(r> = (1 - ?- - S)%(l) + ?-%(2) + SUk(3) (4) where r lies within the triangular element k, its local coordinates being (r, s). The three nodes at the cor- ners of this surface element are characterized by the values of the charge density, oK1), and ak(3). Once a(r) is defined as a continuous function on the molecular surface, the modification of the solute Hamiltonian can be obtained from the corresponding electrostatic potential:

(5)

The surface integration is performed by a Gauss- Radau quadrature on the triangular patches.15

Excited States

The computational procedures outlined above are suitable for evaluating the solvent effect of the molecular ground state but have to be modified for the determination of the solvent effect on electronic excitation~?,’~ To this end, the total ground-state polarization charge is partitioned into an orientational part, which remains unchanged during a vertical electronic excitation, and into an induction contribution, which is assumed to respond instantaneously to the (new) solute charge distribution of the excited state. Therefore, the surface charge density after an excitation is given by

ptot(ex> = @or(pgs> + (6)

To obtain solvent-shifted transition energies, we use the following procedure:

1. Iterative calculation of the ground-state wave function and the ground-state surface charge den-

2. Partitioning of atOt(gs) into an induction and an orientational part.

3. Iterative calculation of mmd(ex) as the induced surface charge density according to a solute charge distribution belonging to the Hamiltonian Ho +

4. Calculation of the excitation energy as the diifer- ence between the ground-state energy of the Ham- iltonian Hgs = Ho + V[a,,(gs)] + V[u,,(gs)] and the energy of the excited state of the Hamiltonian

The key step for obtainiig solvent-shifted excitation energies is therefore the iterative calculation of the quantity ahd(pe.J, the induced surface charge density that corresponds to a solute charge distribution of the Hamiltonian Ho + v[ao,(gs)] + V[crid(ex)]?J7 Here, the FEM procedure offers an additional advantage because the coefficient matrix K, set up already in the ground-state calculation, can be reused without alteration.

sity ato,(gs).

V[aor(gs)l + V[aind(ex)l-

Hex = HO + v(‘or(gs)l + V[mid(ex>l.

COMPUTATIONAL DETAJLS

All electronic structure calculations have been performed using the semiempirical INDO method; the INDO/S parametrization was employed for the calculation of excitation energies?OJ1 The additional modules necessary for the calculation of the solvent effect were written in such a form as to exploit the vectorization capabilities of a CONVEX C210 computer.

A series of test molecules (see Fig. 1) was chosen, mostly with applications toward photo-induced charge transfer processes in mind. The molecular

OH

Figure 1. p-Nitroanilie (l), p-amino-p’-nitro-diphenyl ( Z ) , p-amino-p’-nitro-diphenylethyne (3), and a large ET compound (4).


geometries of amino-nitro compounds 1-3 were taken from the 1iterat~u-e~~; that of compound 4 was determined by a molecular mechanics cal~ulation?~ The charge distribution of the solute is represented by a set of atomic point charges derived from a Mul- liken population analysis. This approximation is in the spirit of the INDO method and may be justified as long as one has to accept the more fundamental deficiencies of a purely electrostatic model.'7z2

Within the INDO approach, the calculation of the matrix elements of V[a](r) involves only one- and two-center integrals. The geometry-dependent part of each integral is calculated only once and stored to reduce the computational effort. During each self- consistency iteration, this quantity has to be multi- plied by the appropriate value of the corresponding surface point charge (MST approach) or the value of the charge density at the surface nodes (FE method), respectively.

In the MST approach, the determination of the mutual polarization of the surface point charges involves the calculation of the mutual distances between them. This distance matrix is calculated once and then stored on a file. Its size scales with N:, and thus it is responsible for the main part of the background files needed, especially for larger numbers of point charges. Further, it has to be read during each iteration cycle. This results in a substantial I/O overhead and causes an unfavorable relation between the CPU time needed for the user activity and that for system activities. In the FE method, setting up the coefficient matrix K requires a considerable amount of CPU time. Therefore, it is calculated only once during the determination of the ground state system matrix (1 - f(q,)K} and stored on file for later use in the calculation of the system matrix for the excited states, (1 - f(eg)K). Its size scales with iV; and (except for the integral file) determines the required background file space in the FE method.

Due to numerical errors, the integral of the polarization charge density u(r) over the total surface differs from its theoretical value (zero for neutral molecules). This entails fluctuations of the results with increasing node density, preventing a well-be- haved convergence. By scaling the positive and neg-

ative contributions to the surface charge density sep- arately such that the changes in the sum of all positive contributions equals that of all negative ones! these fluctuations were reduced by about one order of magnitude, thus resulting in a satisfactory convergence.

The radii for the atomic spheres that define the solute cavity were taken as van der Waals radii scaled by the uniform factor off, = 1.2. In previous investigations, this choice was shown to give reasonable agreement with experimental data.'7,223" This may be necessary to compensate for the overestimation of the solute polarization in cavity models that assume a sudden spatial change of the dielectric ~ o n s t a n t ? ~ , ~ ~ , ~ ~ The radius of the solvent probe molecule necessary for the definition of the Connolly surface was chosen to be 1.5 A. To maximize the solvent effect and any inaccuracy due to the discretization of the surface charge density, we used water as the dielectric continuum throughout this study (E,, = 80.0, E , = T Z ~ = 1.332). Inside the cavity, a dielectric constant of E = 1.0 was assumed. With these values, the ground-state calculations converged after at most nine charge density iterations and the results for the excited states stabilized after four to five iterations (depending upon the amount of charge redistribution).

RESULTS

We have chosen a series of related amino-nitro compounds 1-3 that exhibit both "normal" and charge transfer (CT) excitations and also allow the investigation of the scaling of both methods. Further, we present results for a large molecule (4) that exhibits a photo-induced electron tran~fer.3~

p-Nitroaniline

In Table I, we compare the results of INDO/S-CI calculations on p-nitroaniline (1) for different input values d, of the node density obtained by two different implementations of the MST procedure. To characterize the triangulation, we give the number

Table I. Comparison of the MST method for flat and curved surface patches for p-nitroaniline (see text): number of surface elements N,, total cavity surface area (in A2), ground-state depression AE*NDo (in ev), and the excitation energy of the CT transition vCT (in cm-l) for different values of the input node density d, ( i n k 2 ) .

MST Flat triangles MST: Curved triangles

vCT Area U l N D O Area U ' N D O VCT dn Ne

0.5 302 149.327 1.0694 22,569 164.772 0.6077 24,222 1.0 420 157.628 0.8022 23,428 164.736 0.6075 24,182 2.0 640 160.977 0.6801 23,832 164.725 0.6080 24,148 5.0 1508 163.221 0.6459 23,958 164.720 0.6215 24,058 7.0 2036 163.660 0.6400 23,978 164.720 0.6233 24,048

10.0 2784 164.051 0.6343 23,999 164.720 0.6241 24,040 The calculated value for the CT transition in vacuo is 29,722 cm-'.


of triangular surface elements N, and the total area of the cavity surface, calculated as the sum over the areas of the individual patches. We display the solvent induced depression AFNDo of the ground-state energy as well as the excitation energy of the lowest- lying CT transition, vCT.

In the straightforward implementation of the MST method we used in the past,1722 the given N, surface nodes are connected by flat triangles (MST-IT). This procedure implies a rather coarse description of the cavity surface where the cavity surface area increases in a monotonic fashion with increasing values d, of the node density. The calculated solvent effects on the depression of the ground-state energy and on the solvatochromic shift of the CT excitation energy decrease in size. This behavior may be rationalized by reference to the shape of the actual cavity, which is bounded byflat triangles. As a consequence, the distances between the surface point charges and the solute charge density, represented by effective atomic charges, are underesti- mated, which results in a overestimation of the solvent effect. For input node densities larger than 5.0 A-', reasonable convergence of all observables is obtained. Taking the limiting values of the improved method as a criterion (see below), we estimate the error in the excitation energies to less than 60 cm-'. However, for the three highest densities investigated the rate of convergence is poor: The values for the various observables decrease about linearly with increasing node density. Therefore, the limiting values differ significantly from those obtained for the highest node density of 10.0 k'. This result for the observables may be anticipated also by inspection of the calculated surface area, which, for a node density of 10.0 k2, is still notably lower than the value calculated via curvilinear triangles (164.05 vs. 164.72 A">. From this criterion, one clearly anticipates that higher node densities in the simple MST-IT method are required to obtain satisfactory convergence of the observables.

A drastic improvement over this simplistic implementation of the MST method is achieved when the cavity surface is described by curved triangles. This leads to a "correct" distance of the surface point charges from the atomic centers and better values

for the weighting factors ASj in the calculation of the point charges [see eq. (l)]. The results calculated by this more sophisticated variant of the MST method exhibit an excellent convergence of the surface area. This quantity is practically constant for node densities above 5.0 A-2. Concurrently, the observables show only minor changes for a higher number of surface elements. The variations of the ground-state depression AEINDo are on the order of 10 meV; the energy of the CT transition is constant within 20 cm-'. These findings clearly demonstrate that the use of curved triangles yields a far better convergence of the observables with increasing node density than the previous version employing flat surface patches. Therefore, in the following we employ only this improved version of the MST model.

In Table 11, we compare the results for different input values of the node density d, obtained by both the (improved) MST and FEM procedures. For a given value of d,, both methods are based upon an identical triangulation of the cavity surface; the number of surface elements N, and the total area of the cavity surface displayed in Table I1 are therefore identical for both procedures. We display the ground- state energy depression MINDo and the ground-state dipole moment p, as well as two excitation energies, that of the lowest excited state without charge transfer character, vl, and that of the lowest CT transition,

For the MST method, the dipole moment as well as the excitation energy vl show the same trends as the ground-state depression and vCT discussed above. In general, a monotonic increase of the solvent effect is found. The values obtained for node densities larger than 5.0 Hi-' are converged to an acceptable degree. The ground-state depression varies by about 1 meV, the dipole moment p by about 0.01 Debye, the energy of the lowest transition changes by less than 10 cm-', and that of the CT transition by less than 20 cm-'. Even with the lowest node density investigated (d, = 0.5 k', corresponding to only 302 surface elements), reasonable results are calculated.

The results calculated by the FE method show an excellent convergence as well. However, unlike in the MST method, no monotonic behavior of the sol-

vCT.

Table 11. Results of the MST and FE methods for p-nitroaniline: number of surface elements N,, total cavity surface area (in A2), ground-state depression A E I N D 0 (in el'), ground-state dipole p (in Debye), and the excitation energies of the lowest transition y as well as of the CT transition vCT (in cm-') for different values of the input node density dn (ink').

MST FEM

dn Nt? Area U l N D O 1FL Vl vCT CL v1 vCT U l N D O

0.5 302 164.772 0.6077 10.093 22,735 24,222 0.6498 10.313 22,829 23,794 1.0 420 164.736 0.6075 10.104 22,734 24,182 0.6339 10.262 22,798 23,939 2.0 640 164.725 0.6080 10.110 22,740 24,148 0.6278 10.237 22,783 24,028 5.0 1508 164.720 0.6215 10.156 22,779 24,058 0.6254 10.182 22,783 24,062 7.0 2036 164.720 0.6233 10.162 22,783 24,048 0.6260 10.182 22,785 24,052

10.0 2784 164.720 0.6241 10.165 22,785 24,040 0.6259 10.176 22,786 24,058


vent effect is found; for intermediate values of the node density, a slight but distinct minimum occurs. Its origin cannot be traced in a definite way but is probably connected to the error of the linear interpolation of the surface charge density in the FEM method. Depending upon the triangulation, this interpolation represents the actual values of polarization charge density u(r) to a differing quality. How- ever, for all densities investigated by the FEM procedure the error in the observables is of a moderate size. For a density of 5.0 the results are converged within narrow error margins. The error in the ground-state depression is less than 1 meV while the energy of the lowest transition is practically constant for densities greater than 2.0 k2. The value of the CT transition-which is connected with a significant rearrangement of the solute charge density-is not stabilized for values of the node density below 5.0 Also, the ground-state dipole moment turns out to be a sensitive convergence criterion. Further, the differences in the various calculated quantities obtained for the densities 7.0 and 10.0 k2 seem to indicate the numerical limit of the obtain- able accuracy.

A comparison of the results shown in Table I1 provides strong evidence that the MST and FE methods converge to the same limits. Although for the highest node densities investigated the observables still show some differences (2 meV for AE1IND0, 0.01 Debye for p, and 20 cm-' for vCT), these discrep- ancies are small. They may be ascribed both to the fact that the results may not yet be completely converged and to the inherent numerical accuracy.

A further important criterion for the effectiveness of the two methods-besides the intrinsic convergence behavior with increasing node density-is the computation time and the size of the temporary background files. In Table 111, we display the computational (CPU plus system) time necessary for the ground state and for one excited state. We also show the total computational time as a function of the node density and the time needed for the surface triangulation, as well as the total size of the temporary files. Further, we list the time necessary in

the FE method for the evaluation of the coefficient matrix K and initial calculation of the interaction integrals; both comprise a significant part of the total time required for the ground-state calculation due to the computational economy of the INDO electronic structure method. Inspection of Table 111 shows that for low node densities both methods require about the same computation time. However, for node densities higher than 2.0 A-2 the FE method becomes more advantageous, the differences in computation time increasing with the node density. For d, = 10.0

the FE method is almost twice as fast as the MST method. However, compared to an INDO calculation in vacuo, which takes 12 s on a CONVEX 210 computer, the inclusion of the solvent effect requires significantly longer computation times in both methods. This is partly due to the solute-solvent self- consistent procedure.'722 Various passes through the INDO routines account for about 200 s of the total computation time, a substantial fraction, especially for low node densities.

The FE method offers additional advantages in the reduced size of the background files. Inspection of Table I11 reveals that in the MST method the file size increases drastically with increasing number of surface elements (see the previous section).

p -Amino-p '-Nitro-Diphenyl

The next member in the series of amino-nitro compounds is p-amino-p'-nitro-diphenyl(2) (see Fig. 1). The results obtained by the MST and FE methods are collected in Table IV. Although this molecule is considerably larger than p-nitroaniline (l), satisfactory convergence is achieved with both techniques. Again, the ground-state dipole moment and the excitation energy of the CT state provide sensitive cri- teria for the convergence. Up to a node density d, of 5.0 A-2, significant changes of these observables are found. For the FE method, the ground-state depression AElNDo as well as the energy v1 of the non-CT transition show excellent convergence even for the node density 1.0 kz. In contrast to this satisfactory behavior of the FE method, all results for

Table 111. Timing results (CPU + system in s) for p-nitroaniline: time required on a CONVEX 210 to complete the triangulation (TRI), ground-state calculation (GS), calculation of one excited state (EX), and total time for the ground state and the three lowest excitations (total).

MST FEM dn TRI FS GS EX Total FS Matrix Integ. GS EX Total 0.5 3.5 1.6 24 70 236 1.2 1.9 2.9 31 69 237 1 .o 4.7 2.5 27 69 233 1.5 3.5 4.1 37 69 247 2 .O 8.0 4.6 42 74 260 2.2 7.6 6.1 52 71 270 5.0 21 20.3 134 110 452 6.7 40 14 133 83 395 7.0 34 35.5 255 163 713 11.0 72 20 203 90 498

10.0 57 64.5 448 246 1126 18.7 133 27 327 103 676 The time for the calculation of the coefficient matrix K (matrix) and the integral setup (integ.) in the FJ3 method is

also shown, as is the total size FS of the background files (in MB).


Table IV. Results of the MST and FE methods for p-amino-p'-nitro-diphenyl(2).

MST FEM

EL Y vCT EL Vl VC'T U l N D O Area U l N D O dn Ne

0.5 444 239.761 0.5237 10.159 22,494 22,085 0.5420 10.290 22,537 21,626 1 .o 594 239.720 0.5214 10.160 22,489 22,057 0.5352 10.265 22,524 21,777 2.0 930 239.706 0.5175 10.153 22,479 22,045 0.5349 10.289 22,526 21,800 5.0 2146 239.697 0.5308 10.195 22,520 21,907 0.5317 10.211 22,519 21,922 7.0 2850 239.697 0.5326 10.198 22,526 21,894 0.5343 10.214 22.527 21,900

Number of surface elements N,, total cavity surface area (in d;z), ground-state depression (in ev), ground-state dipole EL (in Debye), and the excitation energies of the lowest transition v1 as well as the CT transition vCT (in cm-') for different values of the input node density dn (in k2).

the MST method converge only at d, = 5.0 k2 and higher. It is gratlfying to note that the results obtained by the two different methods exhibit only negligible differences. The lowering &INDo of the ground state and the transition energies are essentially equal; only the values for the ground-state dipole moment differ (by about 0.015 Debye).

The corresponding computation times are sum- marized in Table V. As found already with p-nitroaniline, for low node densities the computational effort for the two methods is about equal, the FEM becom- ing more advantageous for node densities above 2.0 k2. We note that the multiple INDO passes (about 400 s) comprise a significant part of the total computation time, especially for the lower values of the node density. Also shown in Table V is the total size of the background files, which exhibit the same trends as noted already for p-nitroaniline.

p -Amino -p '-Nitro -Diphenyle thyne

To increase the distance between the donor and acceptor units, acetylene units can be inserted between the two phenyl rings of 2. We investigated the small- est member of this series, p-amino-p'-nitro-diphenylethyne (3) (see Fig. 1).

A comparison of the two cavity methods applied to 3 is shown in Table VI. The ground-state depression calculated by FEM is for densities greater than 2.0 rather stable; the values fluctuate by about 1 meV. This behavior may again be ascribed to the error in the interpolation of the surface charge density. On the other hand, the MST ground-state depression increases monotonically with increasing node density, the value for d, = 7.0 A-2 lying in the region spanned by the FEM results. Here again, the dipole moment seems to be particularly sensitive. With both meth-

Table V. Timing results (CPU + system in s) for p-amino-p'-nitro-diphenyl (2): time required on a CONVEX 210 to complete the triangulation (TRI), ground-state calculation (GS), calculation of the CT state (CT), and total time for the ground state and the three lowest excitations (total).

MST FEM d, TRI Fs GS CT Total Fs Matrix Integ. GS CT Total

0.5 6.7 3.0 54 124 453 1.9 3.9 6.9 66 12 1 460 1 .o 7.8 4.5 63 119 443 2.9 6.6 9.2 75 118 453 2.0 15 9.1 88 130 509 3.8 16 14 106 122 502 5.0 29 40.4 291 220 967 12.7 80 33 247 149 732 7.0 48 69.0 472 301 1379 20.4 140 44 367 163 912

The time for the calculation of the coefficient matrix K (matrix) and the integral setup (integ.) in the FE method is also shown, as is the total size FS of the background files (in MB).

Table VI. Comparison of the MST and FE methods for p-amino-p'-nitro-diphenylethyne (3): number of surface elements N,, ground-state depression UtNDO (in eV), ground-state dipole (in Debye), and excitation energy of the CT transition vcT (in cm-') for different values of the input node density d, (in A-2).

MST FEM

dna Ne EL 4CT) EL GJ3 U N D O U I N D O

1 .o 1034 0.5131 10.271 22,500 0.5298 10.377 22,214 2.0 1098 0.5103 10.265 22,490 0.5261 10.399 22,235 5.0 2488 0.5235 10.305 22,352 0.5244 10.321 22,355 7.0 3318 0.5251 10.307 22,342 0.5263 10.320 22,347

Total surface area: 280.39 A2.


Table VII. Timing results (CPU + system in s) for 3: time required on a CONVEX 210 to complete the ground- state calculation (GS), calculation of the CT state (CT), and total time for the ground state and the three lowest excitations (total).

MST FEM

d, GS CT Total Matrix Intec. GS CT Total 1.0 83 112 433 8.7 12 99 108 437 2.0 119 126 524 21 19 137 121 511 5.0 389 246 1137 107 43 318 140 789 7.0 625 357 1690 198 57 483 159 1035

The time for the calculation of the coefficient matrix K (matrix) and the integral setup (integ.) in the FE method is also shown.

odsb p does not converge for a node density below 5.0 A-', for which a distinct difference of 0.015 Debye between the MST and FEM results is found. Also shown are the results for the lowest CT transition energy. The calculated values are stable for densities greater than 5.0 for both methods, with differences occurring below the expected error margin. The corresponding timing results in Table VII exhibit similar features as found already with the smaller systems.

Larger Systems

To demonstrate the power of the methods presented, we apply them to a large donor-acceptor molecule that has been synthesized re~ently.3~ It consists of a substituted naphthalene as donor unit and a 1,l-di- cyano-ethylene as acceptor group, which are separated by a rigid cT-bridge (4, see Fig. l).

As can be seen from Table VIII, the trend of the results is similar to the one encountered with the amino-nitro-diphenyl compounds. The results of the MST method show a monotonically increasing ground-state depression and dipole moment with increasing node density. However, up to d, = 5.0 no convergence is obtained; from density 2.0-5.0 A-', the ground-state depression varies by 0.03 eV, still differing significantly from the final result obtained by the FEM. However, we were not able to obtain converged results for node densities greater than 5.0 A-2 with the present version of the MST procedure. This may indicate an inherent limitation of the MST method as implemented recently. With the FEM technique, the error of the ground-state

depression is less than 0.02 eV for node densities greater than 1.0 A-', dropping below 1 meV for densities greater than 5.0 A-2. From this aspect, a FEM calculation with an input node density of 2.0 A-z provides results of an accuracy comparable to that of a MST calculation with a density of 5.0 A-2. How- ever, it needs only about one third the computation time. Again, for higher node densities the required calculation time for FEM is much shorter than that required by the MST method. In addition, the file sizes for FEM increase much slower than those for the MST method.

These results demonstrate that even for a molecule consisting of 45 atoms, with an edge-to-edge extension of 11.5 A, good convergence of a self-consistent treatment of the electrostatic solvent effects is possible within the FEM framework, even for a still reasonable computational resource.

Scaling Behavior

Finally, we discuss the scaling behavior of the two methods. In Figure 2, we show for acetone, p-ni-

n v) Y

i! i=

1 5 0 0 ~

1200

900

600

300

-

-

-

-

or":"--l ' 2 ' ' 3 ' ' 4 ' Ne / 1000

Figure 2. Total computation time (ground and excited states) as a function of the number of surface elements N,. Comparison of the MST (0) and FE methods (A) applied to acetone (dotted line), p-nitroaniline (solid line), and p-amino-p'-nitro-diphenylethyne 3 (broken line).

Table VIII. Comparison of the MST and FE methods for the ET compound 4: number of surface elements N,, ground- state depression B I N D " (in eV), ground-state dipole p (in Debye), size FS of the background files (in Mb), and calculation time for the ground state (in s) on a CONVEX C210 computer for different values of the input node density d, (in k2).

MST FEM dn a N, U I N D O P FS Time &lNDO w Fs Time

0.5 704 1.1028 15.859 6.1 191 1.2315 16.078 2.8 208 1 .o 944 1.1117 15.878 9.8 217 1.2069 16.032 3.9 23 1 2.0 1420 1.1391 15.903 19.8 279 1.1952 15.995 6.9 286 5.0 3282 1.1694 15.932 93.0 777 1.1872 15.955 27.2 609 8.0 4630 1.1867 15.951 50.2 964

"Total surface area: 366.50 A2.


’0° m 600 1 I

; * I : ’1

Ne / 1000 Figure 3. Computation time for the ground state as a function of the number of surface elements N,. Compar- ison of the MST (0) and FE methods (A) applied to acetone (dotted line), p-nitroaniline (solid line), and p-amino- p‘-nitro-diphenylethyne 3 (broken line).

troaniline (l), and p-amino-p’-nitro-diphenylethyne (3) the total computation time (i.e., for ground state and CI calculation) as a function of the number N, of patches used for the description of cavity surface. While the calculation times of both methods are almost equal at low node densities, the MST method becomes increasingly more expensive than the FEM technique. As can be seen from Figures 3 and 4, this is mainly due to a different calculation time for the

300

n

24 200 E i=

loot /.4 1 , I .,..?; ._._... , *.-- I , 1

oo ..... 2 3 4

/.4 1 , I .,..?; ._._... , *.-- I , 1 oo .....

2 3 4

Ne /I 000 Figure 4. Computation time for one excited state as a function of the number of surface elements N,. Compar- ison of the MST (0) and FE methods (A) applied to acetone (dotted line), p-nitroaniline (solid line), and p-amino- p’-nitro-diphenylethyne 3 (broken line).

excited states. In Figure 3, the time required for the calculation of the ground state is displayed. For the three molecules shown, the differences between the two methods are negligible for small numbers of the surface elements; for larger N,, the computation time for MST increases slightly faster than that for FEM, the maximum difference being about 25%. We note that the actual time needed is rather independent of the compound investigated and mainly a function of the number of patches N,. On the other hand, the results for the excited states (see Fig. 4) show a significant difference between MST and FEM for an increasing number of surface elements. While the computation time for the FE method increases about linearly, a much steeper increase is observed for the MST method. These results illustrate the superiority of the FE method for larger molecules when high node densities have to be applied. This is especially so when several excited states have to be calculated; in this case, the bulk of the FE overhead has to be dealt with only once (for the ground state).

CONCLUSIONS

We presented an improved continuum model for the calculation of the electrostatic contribution to solvatochromic shifts. It was implemented in the framework of the INDOIS-CI electronic structure method and includes the interaction of the solute as determined self-consistently from the reaction field in a cavity of essentially arbitrary shape. Compared to the previous implementation of the MST method,6J7 both the improved MST method with curved triangles and the formalism based upon the finite element method of ZauharI5 exhibit faster convergence of the resulting observables with increasing node density on the cavity surface and therefore permit an effective investigation of large molecular systems, where a simplistic discretization technique of the surface charge density fails to produce reliable results. For small node densities, both the MST and FE methods require about the same computational effort. How- ever, for higher node densities that are necessary to obtain reliable results (especially for larger molecules) the finite element method exhibits substantial advantages in both computation time and size of the background files. With this effective method at hand, the problems and unsolved questions of current continuum methods become amenable to a systematic and thorough investigation. Among them are the question of how to properly choose the atomic radii or the scaling factor f,. Moreover, there have been alternative suggestions for the partitioning of the “surface” charge density qtot into an inductive and orientational part.% Finally, we mention that almost all cavity models employed so far include only electrostatic contributions to the solvatochromic effect; the influence of dispersion interaction^'^ as well as


directed short-range interactiohs between solute and s0lvents'~2~@ are often neglected. These problems constitute an interesting challenge for future re- search that may be tackled now that the effective FEM method is at hand.

The authors thank M.C. Zerner for stimulating discus- sions. This work has been supported in part by the Bun- desministerium fiir Forschung and Technologie, Germany, and by the Fonds der Chemischen Industrie.

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electrostatic solvent effects on the electronic structure of ground and excited states of molecules:...

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