improving the efficiency and convergence of geometry optimization with the polarizable continuum...

Improving the Efficiency and Convergence of GeometryOptimization with the Polarizable Continuum Model:

New Energy Gradients and MolecularSurface Tessellation

HUI LI, JAN H. JENSENDepartment of Chemistry, University of Iowa, Iowa City, Iowa 52242

Received 5 February 2004; Accepted 19 April 2004DOI 10.1002/jcc.20072

Published online in Wiley InterScience (www.interscience.wiley.com).

Abstract: New equations are derived and implemented for efficient and accurate computation of solvation energyderivatives for the conductor-like polarizable continuum model (C-PCM) and the isotropic integral equation formalismpolarizable continuum model (IEF-PCM). Two new molecular surface tessellation procedures GEPOL-RT and GE-POL-AS that generate near continuous potential energy surfaces are proposed for PCM geometry optimization. Thecombined use of these new techniques leads to efficient and convergent geometry optimizations with the PCMs.

© 2004 Wiley Periodicals, Inc. J Comput Chem 25: 1449–1462, 2004

Key words: geometry optimization; polarizable continuum model; molecular surface tessellation

Introduction

Continuum solvation models are important computational meth-ods for theoretical studies of condensed phase chemistry.1– 4 Ina continuum model the solvent is treated as a structurelessdielectric medium or a conductor, instead of a dynamic ensem-ble of molecules, while the solute is represented by a distribu-tion of charges. The polarizable continuum models (the earlierD-PCM5 and the more recent IEF-PCM6), the conductor-likescreening models (COSMO7 and GCOSMO8 or C-PCM9) andthe SS(V)PE10,11 models are continuum solvation models forwhich the charge distribution is calculated using ab initioelectronic structure theory. The solute/solvent boundary, usu-ally described by a set of interlocking atom-centered spheres, isdivided into small surface elements (tesserae). The Poissonequation is then solved for each tessera to yield an apparentsurface charge (ASC) located at the tesserae center. By care-fully parameterizing the solute cavity construction procedurethe continuum solvation models can generally reproduce theexperimental solvation energies to within a few kcal/mol forsmall charged molecules and to within 1 kcal/mol for smallneutral molecules.7,12–15 Furthermore, the analytical first deriv-atives of the molecular energy with respect to atomic coordi-nates, that is, the gradients, have been derived for D-PCM,16,17

IEF-PCM,18 –21 COSMO,7 GCOSMO,22 and C-PCM,9,23 andhave been applied to molecular geometry optimization in thesolution phase.

However, geometry optimizations with these boundary-ele-ment ASC continuum models may suffer convergence problemssuch as slow convergence, no convergence, or convergence tohigher energetic conformations. We have encountered all of theseproblems in geometry optimizations using the D-PCM, IEF-PCM,and C-PCM methods and the GEPOL-GB24–27 tessellation proce-dure. The chance of encountering problematic optimization casesis quite high, especially for large and flexible molecules.

The cause of the problem is that the potential energy surface(PES) of a molecule treated with PCM is discrete, due to the finiteboundary element approximation used to determine the pointASCs.28–31 Using the GEPOL-GB24–27 tessellation procedure thenumber, shapes, and areas of the tesserae change discontinuouslyas the molecular geometry changes. Furthermore, GEPOL-GBgenerates many boundary tesserae between interlocking spheres sothat the distance between the center points of two boundarytesserae on two neighboring spheres may be too short compared totheir sizes. This can lead to electrostatic interactions between thetesserae that are too strong and can cause instabilities in theiterative algorithms that determine the ASCs. During a geometryoptimization, appearance and disappearance of such boundarytesserae can cause especially large fluctuations in energy and

Correspondence to: J. H. Jensen; e-mail: [email protected]

Contract/grant sponsor: NSF; Contract/grant numbers: CHE 9974502and MCB 0209941

© 2004 Wiley Periodicals, Inc.

gradient as the molecular geometry changes. The situation is evenworse when two spheres are about to separate completely or aboutto merge (e.g., when breaking or forming a hydrogen bond),because large tesserae may approach each other arbitrarily close.

In general, these problems can be addressed by using a largenumber of small tesserae. However, a very large number oftesserae are required to generate a near continuous PES, which isimpractical for large molecules. The separating/merging problemhas been addressed by introducing additional spheres that smooththe solute cavity and surface.25,32,33 However, this can lead to asignificant increase in the number of boundary tesserae.

Other solutions have been proposed to address these generalproblems. York and Karplus30 show that using a Gaussian chargedistribution instead of a point charge for each boundary element cangenerate a relatively smooth potential energy surface (PES). Further,using a continuous switching function to treat the appearance anddisappearance of tesserae leads to a rigorously smooth PES.30 Senn etal.31 implemented a very similar approach for Car–Parrinello ab initiomolecular dynamics. However, none of the two studies have beentested for QM molecular geometry optimizations.

This article presents two novel tessellation procedures thatensure the distance between any pair of tesserae is large enough(compared to their sizes) by discarding or scaling the boundarytesserae. In one procedure, boundary tesserae and nearby tesseraethat may become boundary tesserae during the geometry optimi-zation are subdivided into smaller tesserae. This procedure leads toa significant increase in the number of tesserae and, hence, com-putational cost; and is intended mainly as a benchmark againstwhich other methods can be compared. In the other procedure, theareas of boundary tesserae are reduced by an empirically obtainedcriterion. These two tessellation procedures are able to generatePESs that are more continuous than those generated by the GE-POL-GB procedure.

For molecules with a large number of tesserae, the computationof the analytical IEF-PCM gradients involves large-scale matrixinversion or multiplication, which are costly in both core memoryand CPU time. Approximations must be made before IEF-PCMcan be practically applied for geometry optimizations of largemolecules. In this study approximate IEF-PCM gradients are de-rived and applied for geometry optimization of large and flexiblemolecules, at very low computational costs.

This article is organized as follows. First, after a review of theIEF-PCM and C-PCM theories, we present the approximate IEF-PCM/C-PCM gradients suitable for large systems and two newtessellation procedures that generate near continuous PESs. Sec-ond, we describe the general computational methodology used inthis study. Third, we compare our approximate IEF-PCM gradientsand tessellation procedures with the conventional ones, and applythem to molecular geometry optimizations. Finally, we concludeand discuss future directions.

Theory

Polarizable Continuum Model

In the polarizable continuum model (PCM) the solute molecule isplaced in the bulk solvent described as a polarizable continuum

with a dielectric constant �. The cavity that the solute moleculeoccupies in the bulk solvent can be defined in varieties of ways, ofwhich the most popular one is to use interlocking spheres centeredat atoms or atomic groups. The surface of the cavity is theboundary between the solute and solvent. In the PCM the apparentsurface charge (ASC) method is used to describe the electrostaticinteraction between the solute and the bulk solvent. To numericallysolve the electrostatic boundary equation the continuous chargedistribution on the boundary surface is divided into a set of pointcharges at a finite number of boundary surface elements, calledtesserae. The resulting vector of ASCs, q, are obtained by solvingthe matrix equation,

Cq � g (1)

where the vector g is a function of the solute electrostatic potentialvector, V, and C is a geometric matrix. Both g and C have differentforms for different tessellation methods and different PCM for-malisms. Based on the usual GEPOL24–27 (triangular) cavity con-struction and surface tessellation procedure, the C and g elementsfor the isotropic integral equation formalism PCM (IEF-PCM)6

are:

C � �A2

� D��1�� 1

� � 1

A2

� D�A�1SA�1

g � �V (2)

with

Aii � ai, Aij � 0 (3)

Sii � 1.07ai�4�ai, Sij �aiaj

�r�i � r�j�(4)

Dii � �1.07ai�4�ai

8�Ri, Dij �

aiaj�r�i � r�j� � n̂j

4��r�i � r�j�3(5)

where ri, ai, n̂i, and Ri are, respectively, the center, area, orthog-onal unitary vector, and the sphere radius for tessera i.

For solvents with large dielectric constants, the conductor-likescreening models (COSMO7 and GCOSMO8 or C-PCM9) can beused. The C-PCM, which follows the PCM style of implementa-tion, has the C and g elements:

Cii � 1.07�4�

ai, Cij �

1

�r�i � r�j�(6)

g � �� 1

�V

In both IEF-PCM and C-PCM, the electrostatic interactionbetween the solute and the bulk solvent, referred as PCM solvationenergy hereafter, is:

1450 Li and Jensen • Vol. 25, No. 12 • Journal of Computational Chemistry

G �1

2VT � q (7)

Both IEF-PCM and C-PCM have been implemented in theGAMESS34 program for RHF, ROHF, and UHF type SCF meth-ods and the corresponding density functional theory methods, aswell as the GVB and MCSCF methods.

The first derivatives of the IEF-PCM and C-PCM solvationenergies with respect to an atomic coordinate x are respectively:

Gx � �VT�xq �1

2qTCxq (8)

and

Gx � �VT�xq ��

� � 1�1

2qTCxq (9)

To obtain eq. (8) we have used the approximation that C [cf. eq.(2)] is symmetric, as done previously by Barone and cowork-ers.20,35 This approximation leads to considerably simpler equa-tions for the gradient, and we demonstrate below that the resultinggradient is sufficiently accurate for geometry optimizations.

The derivative of electrostatic potential with respect to theatomic coordinate x, Vx, is evaluated for each tessera. For IEF-PCM, the Cx expands as:

Cx � ��A2

� D��1�� 1

� � 1

A2

� D�A�1SA�1�x

� ��A2

� D��1

� �A2

� D�x�A2

� D��1�� 1

� � 1

A2

� D�A�1SA�1

� �A2

� D��1�� 1

� � 1

A2

� D��x

A�1SA�1 � �A2

� D��1

� �� 1

� � 1

A2

� D��A�1�xSA�1 � �A2

� D��1

� �� 1

� � 1

A2

� D�A�1SxA�1 � �A2

� D��1

� �� 1

� � 1

A2

� D�A�1S�A�1�x (10)

where the derivatives of the A, D, and S matrices can be obtainedby differentiating eqs. (3), (4), and (5), and finally using thederivatives of the areas, orthogonal unitary vectors, and centers ofthe tesserae. For C-PCM, the Cx elements are much simpler,

Ciix � �1.07�4�/ai�

x � �1.07�4�

2�ai�3/ 2 �ai�

x (11)

Cijx � ��r�i � r�j��1�x � � �r�i � r�j��3�r�i � r�j� � �r�i

x � r�jx� (12)

and can be obtained using the derivatives of the areas and centersof the tesserae.

The analytical derivatives of the areas, orthogonal unitary vec-tors, and centers of the tesserae with respect to atomic coordinatesare not available for some cavity construction schemes and surfacetessellation procedures, such as the isodensity–contour–sur-face36–38 and DEFPOL.39,40 The GEPOL-GB33 procedure canprovide these geometric derivatives analytically and allow foranalytical PCM gradient calculations. With these geometric deriv-atives, both the Cx for IEF-PCM and C-PCM can be evaluatedexactly through eqs. (3), (4), (5), (10), (11), and (12). However, thecomputation of Cx for IEF-PCM is very expensive due to matrixinversion and extensive matrix multiplication, and thus is notpractical for molecules with large number of tesserae. A recentlymodified version of IEF-PCM,41 which uses symmetrical interac-tion “weights” instead of apparent surface charges, simplifies theCx evaluation a little, but still needs expensive matrix multiplica-tion.

For solvents with high dielectric constants, C-PCM is a goodsubstitute for IEF-PCM and the Cx elements for C-PCM can easilybe computed through eqs. (11) and (12). For solvents with lowdielectric constants IEF-PCM is required for accurate results.

An alternative to the exact IEF-PCM gradient (which wasimplemented very recently41) is to use the universal Ux(q) approx-imation:18–21

Gx � �VT�Fixx q � Ux�q� (13)

where the first term is evaluated using the fixed-cavity approxi-mation:

GFixx � �VT�Fix

x q (14)

It is important to point out that the fixed-cavity gradient (VT)Fixx q

is not the same as the (VT)xq in (8) and (9), where it is derivedunder the mobile-cavity condition. Both the Ux(q) and the fixed-cavity eqs. (13) and (14) have been implemented to GAMESS forIEF-PCM and C-PCM gradient computations. However, as shownlater in this article, neither the fixed-cavity nor the Ux(q) approx-imation is accurate enough for geometry optimizations, even forhigher dielectric solvents.

In summary, one general difficulty encountered for PCM ge-ometry optimization is the evaluation of geometric derivatives forthe tesserae, which are not available for some cavity constructionand tessellation procedures that are therefore limited to PCMenergy evaluations. The particular difficulty in computing theanalytical IEF-PCM gradients is that the explicit evaluation of Cx

is very complicated and expensive due to extensive matrix multi-plications and inversions, and thus not practical for large mole-cules. The presently available approximations such as the Ux(q)and the fixed-cavity ones are computationally efficient but are notsufficiently accurate. In the next section, we present approximateIEF-PCM gradients that are accurate and do not involve matrixstorage, inversion, or multiplication.

Approximate Solvation Energy Gradients

In this section we introduce a conceptually new approach to thederivation of PCM gradients. This approach leads to significantly

Geometry Optimization with the Polarizable Continuum Model 1451

simpler expressions for the IEF-PCM gradient, making accurategeometry optimizations possible for systems with large number oftesserae.

The Analytical Constant Tesserae Number Gradients

The conventional analytical gradient evaluation, which include the(ai)

x, (n̂i)x and (r�i)

x terms, are derived under the condition that thetotal number of tesserae stays constant upon an infinitely small(virtual) geometry change. Therefore, we refer to this conventionalapproach as the constant tessera number (CTN) gradients hereaf-ter. We note that the use of these gradients to predict a finitegeometrical change implicitly assumes that the number of tesseraestays constant, and that assumption is often wrong.

The Variable Tesserae Number Approximation

We take a conceptually different approach by assuming that thenumber of tesserae is not a constant (tesserae may appear anddisappear), but that the shapes of all tesserae stay fixed when thegeometry is changed, so that (ai)

x � (n̂i)x � 0. As shown below,

this leads to a very simple IEF-PCM gradient equation that can besatisfactorily used for geometry optimization. The gradients com-puted with this assumption are therefore referred to as the variabletessera number (VTN) gradients hereafter.

Although conceptually distinct, the VTN gradients differ fromthe analytical CTN gradients only in how the geometric derivativesof the boundary tesserae belonging (group S) or connected (groupT) to the moving sphere (i.e., the sphere centered at the coordinatex) are computed. For the analytical CTN gradients,

�ai�x 0, �n̂i�

x 0, 0 � �r�i�x � 1 (15)

are used for these tesserae. While for the VTN gradients,

�ai�x � 0, �n̂i�

x � 0, �r�i�x � 1 (16)

�ai�x � 0, �n̂i�

x � 0, �r�i�x � 0 (17)

are used for the tesserae belonging (group S) and connected (groupT) to the moving sphere, respectively.

In both CTN and VTN gradients, the geometric derivatives forthe nonboundary tesserae belonging (group U) to the movingsphere and the remaining tesserae (group V) are, respectively:

�ai�x � 0, �n̂i�

x � 0, �r�i�x � 1 (18)

�ai�x � 0, �n̂i�

x � 0, �r�i�x � 0 (19)

Just like for the CTN gradients, the VTN gradients are derivedassuming an infinitesimal displacement. Thus, new tesserae willhave an infinitely small area and an infinitely small contribution tothe energy, and thus need not be evaluated. In practice, the bound-ary tesserae (groups S and T) are only a minor portion of all thetesserae, and their contributions to the gradients are comparativelysmall. Hence, gradient approximations involving these tesserae [cf.eqs. (16) and (17)] will introduce relatively small errors. When a

higher tessera density is used, the percentage of tesserae at bound-aries becomes smaller and the difference between the VTN andCTN gradients becomes smaller. In the limit of an infinite numberof tesserae, the VTN and CTN gradients converge to the samevalue.

Using the VTN geometric derivative Ax � (A�1)x � 0 tosimplify eq. (10) leads to:

Cx � �A2

� D��1

Dx�A2

� D��1�� 1

� � 1

A2

� D�A�1SA�1

� �A2

� D��1

DxA�1SA�1 � �A2

� D��1

� �� 1

� � 1

A2

� D�A�1SxA�1 (20)

and finally (see appendix for derivation):

Gx � �Vx�Tq �1

2qTA�1SxA�1q �

1

2q2

TA�1SxA�1q

�1

2�� 1�q2

TA�1DxA�1SA�1q2 (21)

where q2 is the second phase induced charge in the two-phaseiterative solution of the IEF-PCM equation,42 and

Diix � Sii

x � 0 (22)

Dijx � ��aiaj/2��r�i � r�j��3�r�i

x � r�jx� � n̂j (23)

Sijx � �aiaj�r�i � r�j��3�r�i � r�j� � �r�i

x � r�jx� (24)

The C-PCM gradients [cf. eq. (9)] are also simplified by usingthe VTN geometric derivatives in that [cf. eq. (11)]:

Ciix � 0 (25)

Truong and Stefanovich22 have used the same assumption for theirGCOSMO gradients.

The VTN eqs. (21) and (9) have been implemented inGAMESS for IEF-PCM and C-PCM gradient computations. Inthese gradient computations two main steps are involved. The firststep is the calculation of the derivatives of solute electrostaticpotential with respect to nuclear displacements for each tessera,that is, the first term in eqs. (21) and (9). The second step is thecalculation of the interactions between apparent surface charges,that is, the remaining terms in eqs. (21) and (9). Step one is themost time-consuming step, while step two involves only pointcharge interactions and is considerably faster. No matrix storage,inversion, or multiplication is required in the VTN gradient com-putations, because all the elements in the matrices S, Sx, Dx (forIEF-PCM) and Cx (for C-PCM) are simple enough to be calculatedwhen needed. The last term in eq. (21) is computed in two steps toavoid matrix multiplication. In the first step the temporary vector


A�1SA�1q2 is computed and stored [recall that A is diagonal, cf.eq. (3)], and used to compute the remaining terms in a second step.

Molecular Surface Tessellation

In this section two new tessellation procedures that generate nearcontinuous PESs are presented.

GEPOL-RT

Near continuous PESs can be obtained by using many very smalltesserae. Because the discontinuity in tessellation occurs at theboundary between the spheres, use of small tesserae is only nec-essary in this region. We have implemented a modified tessellationprocedure termed GEPOL-RT (RT stands for Regular, i.e., near-equilateral Tesserae) that uses a higher density of tesserae nearsphere boundaries. Unlike GEPOL, which uses boundary tesserae,the GEPOL-RT procedure recursively divides each original tesseraacross or close to the boundary into four smaller tesserae until thetessera areas are smaller than a cutoff criterion. The boundarytesserae with areas smaller than the cutoff are discarded, while theremaining tesserae, which are all regular, that is, near-equilateral(but have different sizes), are taken as the final set of tesserae. Thetessera density at the boundary and the total number of tesserae aredetermined by two criteria: (1) the area criterion for stoppingrecursive subdivision and discarding boundary tesserae, and (2) thedistance criterion for defining nonboundary tesserae “close” to theboundary.

Excluding the boundary tesserae leads to an underestimation ofthe actual area of the cavity formed by interlocking spheres,especially when a relatively large criterion is used. However, wewill show that the ASCs and solvation energy are relatively in-sensitive to the total surface area.

The GEPOL-RT method leads to PCM calculations that areessentially converged with respect to the number of tesserae.However, the resulting tessera number can be very large and in thisarticle the GEPOL-RT results are used as benchmark calculationsto validate a more approximate approach described next.

GEPOL-AS

We have implemented another modified procedure termed GE-POL-AS, where the AS stands for Area Scaling. The GEPOL-ASprocedure is identical to GEPOL-GB except that the areas of thetesserae generated by GEPOL-GB are scaled according to theirdistances to neighboring tesserae to ensure that the distance be-tween any pair of tesserae is large enough compared to their areas:

�r�i � r�j� � P1 � P2 � ��ai � �aj�

P1 �1

�48

P2 � �1 � �1 � n̂i � n̂j�4� (26)

If (26) is not satisfied, both ai and aj are scaled by the sameconstant kij (which is always �1):

ai � ai � kij

aj � aj � kij

kij � � �r�i � r�j�P1 � P2 � ��ai � �aj�

� 2

(27)

so that

�r�i � r�j� � P1 � P2 � ��ai � �aj� (28)

According to eqs. (26)–(28), two types of tesserae are subject toscaling. The first type consists of boundary tesserae between twoor more interlocking spheres (small �r�i � r�j�). The second typeconsists of nonboundary tesserae located at two separated spheresthat are close to each other (n̂i � n̂j �1). Thus, GEPOL-AS canhandle sphere separation and merging that may happen during thegeometry optimization of large and flexible molecules, withoutusing additional spheres.

The value of P1 and functional form of P2 were determinedempirically, guided by the following considerations. For twoneighboring nonboundary tesserae i and j (with areas ai and aj)generated by GEPOL-GB on the same sphere, the distance be-tween their center points is approximately

�r�i � r�j� � Pij �1

�5.2��ai � �aj� (29)

However, for two neighboring boundary tesserae on two interlock-ing spheres, their center point distance may be much smaller thanPij. A very short distance causes very strong interactions betweenthe two tesserae and should be prevented. Empirically we foundthat separations below approximately Pij/3,

Pij

3�

1

�48��ai � �aj� (30)

are the most problematic, and that scaling the areas accordingly isan efficient and simple solution.

We have found that this scaling can be insufficient when thedistance between tesserae centers is short because the two spheresare about to merge or have just separated completely. Such tesserapairs are easily identified by the dot products of unit vectorsnormal to the surface (n̂i � n̂j) less than about �0.5. The functionalform of P2 in eq. (26) leads to a sharply increasing scaling as n̂i �n̂j approaches �1. Two and six were also considered for theexponent, but four was found to work best.

Implementation Details

The GEPOL-RT and GEPOL-AS procedures have been imple-mented in the GAMESS34 program. In GEPOL-RT the defaultarea cutoff criterion is 0.01 Å2 and the default distance criterion foreach tessera is the square root of its area. If the distance betweenany one of the vertices of a given tessera and another sphere is lessthan the distance criterion, the tessera is defined as “close” to theboundary and is subject to subdivision. In this implementation,


each sphere can be described with either the GEPOL-GB orGEPOL-AS or GEPOL-RT tessellation procedure and with either60 or 240 or 960 initial tesserae. Different spheres can be describedwith different tessellation procedures and different numbers ofinitial tesserae. These options allow the user to define differenttesserae densities for different parts of a large molecule as neces-sary, for example, when the solute is treated by combinedQM/MM method.

Computational Methodology

The RHF/6-31G(d) level of theory is used for all the calculationsin this study. Both IEF-PCM and C-PCM are used for comparison.In all the PCM calculations, the UAHF14 radii are used for mo-lecular cavity construction without additional spheres (RET �100). For both IEF-PCM and C-PCM the apparent surface chargeswere determined by a semiiterative DIIS procedure42,43 withoutrenormalization (ICOMP � 0).44 Unless otherwise noted, thesolvent is water with � � 78.39. The default number of initialtesserae, 60 per sphere for GEPOL-GB, GEPOL-AS, and GEPOL-RT, and the default area and distance criteria for GEPOL-RT (seeabove), are used unless otherwise indicated. Geometry optimiza-tion is performed in internal coordinates generated by the “auto-matic delocalized coordinates” algorithm to improve the conver-gence.45 The details for specific calculations are reported in thenext section.

All calculations were performed with a locally modified ver-sion of the GAMESS34 program on four-CPU IBM RS/6000 44P270 workstations, an eight-node QS8-2400C QuantumStation andcomputers at the Advanced Biomedical Computing Center(ABCC), SAIC-Frederick, National Cancer Institute at Frederick.The methods described herein will be made publicly available inthe next release of GAMESS.

All calculations presented in this article are performed with thefixed-cavity, Ux(q) and VTN approximate gradients. A directcomparison with the CTN gradients is currently not possible,because area derivatives are not available in the GAMESS pro-gram.

Results and Discussion

Gradients

The molecular energy gradients for acetic acid in water werecomputed with IEF-PCM at the geometry optimized in the gasphase. The IEF-PCM solvation energy gradients were computedwith the fixed-cavity, Ux(q), VTN equations and double numericaldisplacements using a step size of 0.0010 au. The results arepresented in Table 1.

Acetic acid has two carbon, two oxygen, and four hydrogenatoms (Table 1). In the UAHF approach, hydrogen atoms have nospheres assigned. Thus, their PCM solvation energy gradientscomputed by Ux(q) and VTN equations are identical to thosecomputed by the fixed-cavity equation. The numerical gradientsfor these four hydrogen atoms are all within 1 � 10�5 au com-pared to those obtained by the fixed-cavity equation, indicating

that the numerical gradients obtained by double displacement arereliable. We note that the default convergence criterion inGAMESS for geometry optimization is that the maximum andrmsd of gradients be less than 10 � 10�5 and 3.33 � 10�5 au,respectively.

It is clear that the PCM solvation energy gradients computedwith the fixed-cavity approximation have the largest errors com-pared to numerical values. Because the fixed-cavity approximationhas no Cx term [see eq. (12)], using the GEPOL-RT proceduredoes not improve the quality of the gradients. The maximumdeviations from the numerical gradients for GEPOL-GB, GEPOL-AS, and GEPOL-RT are 543 � 10�5 au, 565 � 10�5 au and683 � 10�5 au (for the x coordinates of O4 in Table 1), respec-tively. If only the 12 Cartesian coordinates of the four heavy atomsare considered the rmsd (rmsd12) are 227 � 10�5 au, 233 � 10�5

au, and 272 � 10�5 au, respectively. If all the 24 Cartesiancoordinates are considered the rmsd (rmsd24) are 160 � 10�5 au,164 � 10�5 and 192 � 10�5 au, respectively. These resultsindicate that the fixed-cavity approximation is quite poor and thegeometry found with the fixed-cavity gradients may not be a trueminimum on the PES.

The gradients computed with the Ux(q) approximation aremuch better than those with the fixed-cavity equation. The maxi-mum deviation, rmsd(12) and rmsd(24) are 75 � 10�5, 26 �10�5, and 19 � 10�5 au, respectively for GEPOL-GB, and are71 � 10�5, 24 � 10�5, and 17 � 10�5 au, respectively, forGEPOL-AS, and are 59 � 10�5, 24 � 10�5, and 17 � 10�5 au,respectively, for GEPOL-RT. Improving the tessellation schemecannot improve the accuracy of the gradients computed with theUx(q) approximation. The errors in the Ux(q) gradients are muchhigher than the maximum gradient cutoff (10 � 10�5 au) used inGAMESS for geometry optimizations.

The gradients computed with the VTN approximation are clos-est to the numerical results. The maximum deviation, rmsd(12) andrmsd(24) are 35 � 10�5, 15 � 10�5, and 11 � 10�5 au, respec-tively, for GEPOL-GB, and are 23 � 10�5, 11 � 10�5, and 8 �10�5 au, respectively, for GEPOL-AS, and are 7 � 10�5, 4 �10�5, and 3 � 10�5 au, respectively, for GEPOL-RT. As ex-pected, improving the tessellation improves the quality of thegradients computed with the VTN equation significantly. Theerrors in the VTN gradients are much closer to the convergencecriterion of 10 � 10�5 au.

The computational costs for PCM solvation energy gradientscomputed with the fixed-cavity, the Ux(q) and VTN approximationare found to be practically identical. However, compared to GE-POL-GB and GEPOL-AS, the use of the GEPOL-RT tessellationscheme can increase the computational cost significantly, due tothe large number of tesserae (159 for GB/AS vs. 4269 for RT).

Surface Area in GEPOL-AS and GEPOL-RT

In the GEPOL-RT tessellation procedure, the initial tesserae at theboundary region between the interlocking spheres are recursivelysubdivided until their areas are smaller than a criterion. Theboundary tesserae with areas smaller than this area criterion areexcluded from the final set of tesserae. In GEPOL-AS, the areas ofthe tesserae that are too close to each other are scaled down toavoid strong interactions. Thus, both GEPOL-RT and GEPOL-AS


generate a set of tesserae with total area less than the actual surfacearea of the molecular cavity formed by the interlocking spheres.

Table 2 lists the total number and area of the tesserae generatedfor acetate by GEPOL-GB, GEPOL-AS, and GEPOL-RT. Theactual surface area of the molecular cavity is 94.09 Å2 as calcu-lated by GEPOL-GB. Because of area scaling, the total surfacearea in GEPOL-AS is slightly smaller than that in GEPOL-GB.The area scaling in GEPOL-AS depends on the sizes of thetesserae so that the total area in GEPOL-AS is almost identical tothat in GEPOL-GB when a high density of tesserae is employed.The total tessera area given by GEPOL-RT depends on the areacriterion. When a very small criterion of 0.0001 Å2 is used the totalarea is 93.80 Å2, within 0.3% of the actual area. When a largecutoff 0.04 Å2 is used the total area is underestimated by 6.5%.

Interestingly, the total apparent surface charge and the totalenergy computed for acetate in water are not sensitive to thesurface area (Table 2). For example, using 60 initial tesserae andthe default area cutoff of 0.01 Å2 for GEPOL-RT the IEF-PCMsurface charge is 0.97599, within 0.00030 au of the correspondingGEPOL-GB value. Similarly, the corresponding difference in totalenergies is only 6 � 10�5 au or 0.04 kcal/mol.

The results for C-PCM are very similar to those obtained withIEF-PCM.

Geometry Optimization

Small Molecules

The numbers of steps required to reach the convergence tolerance(10 � 10�5 au) in IEF-PCM geometry optimizations of 10 smallmolecules are presented in Table 3. All searches with the fixed-cavity, Ux(q) and VTN gradients converge in various numbers ofsteps except the one for acetic acid with the Ux(q) gradients andGEPOL-GB tessellation. In general, for a given tessellationmethod, the Ux(q) and VTN gradients deliver much faster con-vergence than the fixed-cavity ones. Although improving the tes-sellation from GEPOL-GB/AS to GEPOL-RT significantly de-creases the number of steps in geometry optimization, it requiresmuch more computation time due to the use of a large number oftesserae. The energy profiles in the IEF-PCM geometry optimiza-tions of acetic acid with various tessellation procedures and gra-dients are shown in Figure 1a (for GEPOL-GB) and Figure 1b (for

Table 1. IEF-PCM Cartesian Gradients (in Units of 10�5 au) with Respect to Nuclear Displacements forAcetic Acid in Water Computed by the Fixed-Cavity, Ux(q) and VTN Equations.

Atom

GEPOL-GB GEPOL-AS GEPOL-RT

Fixed Ux(q) VTN Num. Fixed Ux(q) VTN Num. Fixed Ux(q) VTN Num.

C1 X �112 �142 �158 �154 �112 �142 �158 �154 �115 �145 �163 �164Y �10 1 12 5 �10 1 12 5 �9 1 11 6Z 31 29 24 27 31 29 24 27 35 32 29 23

C2 X 1523 1525 1553 1537 1523 1525 1531 1516 1543 1543 1546 1546Y 519 508 529 518 517 514 513 513 524 521 520 526Z �813 �814 �812 �805 �813 �811 �810 �818 �826 �825 �824 �817Z �813 �814 �812 �805 �813 �811 �810 �818 �826 �825 �824 �817

O3 X �1251 �1053 �1071 �1064 �1251 �1053 �1071 �1064 �1268 �1067 �1078 �1076Y �206 �175 �181 �197 �205 �174 �180 �196 �205 �181 �183 �179Z 461 390 397 408 461 390 397 408 467 401 406 408

O4 X �50 503 488 493 �50 499 511 515 �33 591 650 650Y �604 �1033 �1034 �999 �603 �1021 �1018 �995 �625 �1097 �1141 �1143Z 585 866 963 941 585 883 962 954 591 897 925 923

H5 X �836 �836 �836 �836 �837 �837 �837 �837 �984 �984 �984 �985Y 735 735 735 736 735 735 735 736 856 856 856 856Z �623 �623 �623 �623 �624 �624 �624 �624 �586 �586 �586 �586

H6 X 42 42 42 42 42 42 42 42 45 45 45 45Y �73 �73 �73 �73 �73 �73 �73 �73 �76 �76 �76 �76Z 52 52 52 52 52 52 52 52 58 58 58 59

H7 X �14 �14 �14 �13 �14 �14 �14 �14 �19 �19 �19 �20Y �35 �35 �35 �35 �35 �35 �35 �35 �39 �39 �39 �39Z �50 �50 �50 �50 �50 �50 �50 �50 �53 �53 �53 �53

H8 X �5 �5 �5 �5 �5 �5 �5 �5 2 2 2 3Y 45 45 45 45 45 45 45 45 51 51 51 51Z 50 50 50 51 50 50 50 51 46 46 46 46

Max deviation 543 75 35 565 71 23 683 59 7Rmsd (12) 227 26 15 233 24 11 272 24 4Rmsd (24) 160 19 11 164 17 8 192 17 3

Numerical gradients are computed by double displacements using a step size of 0.0010 au. Rmsd(24) and rmsd(12) referto the root-mean-square deviations from the numerical results for all atoms and all nonhydrogen atoms, respectively.


GEPOL-AS and RT). The corresponding root mean square (rms)gradient profiles are shown in Figure 1c (for GEPOL-GB) andFigure 1d (for GEPOL-AS and RT).

Figures 1a and 1b clearly shows that compared to the con-verged tessellation procedure GEPOL-RT, the GEPOL-GB/ASresult in �0.0005 au higher total energies, presumably due to thelarge repulsion between some boundary tesserae (note that bothGEPOL-GB and GEPOL-AS use boundary tesserae). The energyincreases are almost a constant for different geometries during theoptimization, indicating that there is a level shift of the potentialenergy surface (PES) on going from GEPOL-GB/AS to GEPOL-RT. For a given tessellation procedure, geometry optimizations ofacetic acid using the fixed-cavity gradients lead to a structure thatis �0.0001 au higher in energy than that found using the Ux(q)and VTN gradients, due to the large errors in the fixed-cavity

gradients. The Ux(q) and VTN gradients lead to conformationswith virtually identical energies.

GEPOL-GB may result in very large energy fluctuations duringthe geometry optimization (Fig. 1a). For example, using the Ux(q)gradients, the energy fluctuation can be as large as 0.0016 au or 1kcal/mol. Using the VTN gradients, the energy fluctuation is muchsmaller (0.0003 au or 0.2 kcal/mol). Using the fixed-cavity gradi-ents, the energy fluctuation is very small (0.00002 au or 0.01kcal/mol). GEPOL-GB may also result in very large fluctuations ingradients (Fig. 1c). For example, using the Ux(q) gradients, therms gradient fluctuates and stays at a relatively higher level abovethe convergence tolerance. Using the VTN gradients, the rmsgradient also fluctuates but finally falls below the convergencetolerance. Using the fixed-cavity gradients, the rms gradientchanges smoothly during the optimization and falls below the

Table 2. The Number of Tesserae (NTS), the Surface Area (Å2), the Apparent Surface Charges, ASC, andMolecular Energies, E, (ASC and E both in au) Computed for Acetate in Water Using Various InitialNumbers (60, 240, or 960) of Tesserae in GEPOL-GB and GEPOL-AS, and Various Area Criteria(0.04, 0.01, 0.001, or 0.0001 Å2) in GEPOL-RT.

IEF-PCM C-PCM

NTS Area ASC E ASC E

GB60 156 94.09 0.97569 �227.3467099 0.97554 �227.3468326GB240 551 94.09 0.97622 �227.3469465 0.97610 �227.3470796GB960 2025 94.09 0.97623 �227.3470093 0.97614 �227.3471511AS60 156 94.03 0.97610 �227.3467202 0.97553 �227.3468381AS240 551 94.08 0.97622 �227.3469468 0.97610 �227.3470790AS960 2025 94.09 0.97623 �227.3470093 0.97614 �227.3471511RT0.0400 1680 88.01 0.97441 �227.3460737 0.97460 �227.3462417RT0.0100 3970 91.17 0.97599 �227.3467701 0.97599 �227.3469185RT0.0010 13221 93.14 0.97640 �227.3470281 0.97632 �227.3471712RT0.0001 46438 93.80 0.97642 �227.3470605 0.97635 �227.3472068

The total ASC and total energies are not sensitive to the surface area of the molecular cavity. The gas phase energy is�227.2250685 au.

Table 3. The Numbers of Steps for IEF-PCM Geometry Optimizations in Water Using the GradientsComputed by Fixed-Cavity, Ux(q) and VTN Equations.

GEPOL-GB GEPOL-AS GEPOL-RT

Fixed Ux(q) VTN Fixed Ux(q) VTN Fixed Ux(q) VTN

CH3COOH 20 NC 11 20 20 10 6 4 4CH3COO� 5 5 5 5 5 5 5 5 5CH3NH3

13 6 5 13 6 5 4 4 3CH3NH2 14 14 13 14 14 13 7 7 7Imidazolium 9 9 9 9 9 9 4 4 4Imidazole 10 10 9 10 10 9 4 4 4C6H5OH 14 13 13 14 13 13 5 4 4C6H5O� 5 5 7 5 5 7 4 4 4CH3SH 3 3 3 3 3 3 3 3 3CH3S� 4 4 4 4 4 4 4 4 4

Delocalized internal coordinates are used with the gradient convergence tolerance � 10�4. NC means the geometryoptimization shows no sign of convergence.


convergence tolerance quickly, albeit to a structure with a slightlyhigher energy.

GEPOL-AS and GEPOL-RT do not result in large energy andgradient fluctuations during the optimizations (Fig. 1b and d). Therms gradients change smoothly and fall below the convergencetolerance quickly. It is notable that the GEPOL-AS performs aswell as the converged GEPOL-RT tessellation does, but uses amuch smaller number of tesserae (�160 vs. �4270) and much lesscomputing time (15 times faster for each step).

In short, for these small molecules use of the fixed-cavitygradients can always lead to convergence with various tessellationprocedures. It also seems that the GEPOL-GB tessellation canalways lead to convergence if the VTN gradients are used. How-ever, for large and flexible molecules the fixed-cavity gradientsand GEPOL-GB tessellation are not always sufficient, presumablydue to more boundary tesserae, and a greater chance for largefluctuations in the gradients.

Large Molecules

Table 4 lists the number of steps in C-PCM geometry optimiza-tions performed for 10 large molecules (Fig. 2) in water. Thesemolecules were extracted46 from the X-ray crystal structure (PDBentry 1PPF47) of the protein turkey ovumucoid third domain

(OMTKY3). They have 27 to 49 atoms, of which six to eightatoms of the side chain carboxyl groups are optimized.

Only 3 of the 10 molecules converge when the fixed-cavitygradients and GEPOL-GB with 60 initial tesserae per sphere areused (Table 4). Because the fixed-cavity gradients are insensitiveto tessellation, improving the tessellation quality will not increasethe chance of convergence significantly. For the VTN gradients,only 5 of the 10 molecules converge when GEPOL-GB and 60initial tesserae are used (Table 4). When 240 and 960 initialtesserae are used, the number of converged searches increases to 7.Using GEPOL-AS and GEPOL-RT improves the convergencegreatly. As shown in Table 4, using GEPOL-RT with 240 initialtesserae and area criterion 0.01 Å2, nine are converged. UsingGEPOL-AS, all of them are converged.

The energy profiles for the C-PCM geometry optimizations ofthe acid form of Asp7 with the fixed-cavity gradients and VTNgradients are shown in Figure 3a. The corresponding root meansquare gradient profiles are shown in Figure 3b.

It is clear that the fixed-cavity gradients and GEPOL-GB (60initial tesserae per sphere) tessellation do not yield smooth energyand rms gradient profiles for the optimization (Fig. 3a and b) andshows no sign of convergence. The energy fluctuation is as largeas 0.024 au or 15 kcal/mol.

Figure 1. Energy profiles for IEF-PCM geometry optimizations of acetic acid with (a) GEPOL-GBtessellation and (b) GEPOL-AS/RT tessellations; rms gradient profile in IEF-PCM geometry optimiza-tions of acetic acid with (c) GEPOL-GB tessellation, and (d) GEPOL-AS/RT tessellations. Note thelogarithmic scale for rms gradients, and that a low convergence tolerance of 10�6 au is used comparedto Tables 3 and 4.


Using the VTN gradients and GEPOL-GB (240 initial tesseraeper sphere) leads to large energy and gradient fluctuations, with nosign of convergence. The energy fluctuation is as large as 0.007 auor 4.3 kcal/mol.

Using the VTN gradients and GEPOL-RT (240 initial tesseraeper sphere and 0.01 Å2 as the area criterion), the energy profile isvery smooth for the first 63 steps. However, large energy fluctu-ations occur at steps 64 (0.011 au or 7 kcal/mol) and 69 (0.017 auor 11 kcal/mol), due to separation/merging of two spheres. Thetotal number of tesserae changes from 21,342 to 20,771 to 21,219for the three consecutive steps 63–65, and from 21,247 to 20,785to 21,206 for the three consecutive steps 68–70, clearly indicatingthat two spheres separate and then merge twice. As mentionedbefore, the GEPOL-RT procedure cannot handle this situation. Inthis case, the rms gradient fluctuates and stays above the conver-gence tolerance, showing no sign of convergence. The number oftesserae generated by GEPOL-RT (240 initial tesserae per sphereand 0.01 Å2 as the area criterion) for the acid form of Asp7 is�21,000, much larger than the �1900 generated by GEPOL-GBand AS (240 initial tesserae per sphere). Performing geometryoptimization with GEPOL-RT is very expensive for relativelylarge molecules.

Using the VTN gradients and GEPOL-AS (240 initial tesseraeper sphere) leads to an energy profile that is much smoother thanthe GEPOL-GB and GEPOL-RT cases (Fig. 3a). The largestenergy fluctuation is 0.0024 au or 1.5 kcal/mol. Most importantly,in this case the rms gradient fluctuates within a much smaller range(note the logarithm scale for the rms gradient in Fig. 3b) and finallyfalls below the convergence tolerance.

We note that the acid form Asp7 represents the most challeng-ing case in this study. Figures 3c and d shows the energy and rmsgradient profiles for a more typical case, the base form of Glu43,for which all methods are able to converge the optimization exceptfor the fixed-cavity case (Table 4). It is clear that the fixed-cavitygradients (GEPOL-GB and 60 initial tesserae per sphere) lead to a

structure with significantly higher energy (by �0.002 au or 1.3kcal/mol) than do VTN gradients, and show no sign of conver-gence due to rms gradient fluctuations. The VTN gradients lead tostructures with very similar energies for all the GEPOL-GB/AS/RT tessellations (240 initial tesserae per sphere). However, theGEPOL-GB method leads to large energy fluctuations (as large as0.001 au or 0.7 kcal/mol, Fig. 3c) and rms gradient fluctuations(0.01 au, Fig. 3d) before the optimization tolerance is met. WhenGEPOL-AS and GEPOL-RT are used, no energy fluctuation occursduring the optimizations. The rms gradients fluctuate slightly then fallbelow the gradient tolerance quickly (Fig. 3d). Compared to GEPOL-GB/AS, the GEPOL-RT method generates many more tesserae(�1500 vs. �15,000) and requires much more computing time.

In short, for large and flexible molecules, GEPOL-GB is notsufficient for PCM geometry optimizations due to the appearanceand disappearance of a large number of boundary tesserae betweenspheres. GEPOL-RT can lead to convergence for most cases, butcannot handle the separation or merging of spheres, and is veryexpensive due to the large number of tesserae. GEPOL-AS leads toconvergence for all the molecules studied in this work, with thesame computational cost as the conventional GEPOL-GB ap-proach. The use of GEPOL-AS is recommended for PCM geom-etry optimization.

Conclusions

This article presents several methodological innovations for effi-cient and numerically stable IEF-PCM/C-PCM geometry optimi-zations.

The computation of the exact IEF-PCM gradients is very ex-pensive due to matrix inversion/multiplication and is impracticalfor large molecules. The variable tessera number (VTN) approxi-mate gradients are derived and implemented for both IEF-PCMand C-PCM and applied to relatively large biomolecular geometry

Table 4. The Numbers of Steps for C-PCM Geometry Optimizations in Water Using the GEPOL-GB,GEPOL-AS, and GEPOL-RT Tessellation Procedures.

NAtom NOpt Fixed60

GEPOL-GB

960

GEPOL-AS GEPOL-RT

60 240 60 240 240

Asp7-acid 36 8 NC NC NC 95 137 107 NCAsp7-base 35 7 NC NC NC 58 38 32 33Glu10-acid 28 7 NC 42 41 32 40 37 40Glu10-base 27 6 NC NC 39 40 44 39 42Glu19-acid 49 8 NC 43 36 NC 32 31 29Glu19-base 48 7 62 NC NC NC 33 33 30Asp27-acid 48 7 22 NC 24 24 22 24 24Asp27-base 47 6 31 97 31 NC 16 15 13Glu43-acid 28 7 NC 48 45 45 49 45 45Glu43-base 27 6 NC 38 97 42 67 39 42

The C-PCM solvation energy gradients are computed by the VTN equations. Delocalized internal coordinates are usedwith the gradient convergence tolerance � 10�4 au. NC means the geometry optimization shows no sign ofconvergence.


optimizations (Fig. 2). Compared to the presently available fixed-cavity and Ux(q) approximations, the VTN approximation givesmuch more accurate gradients at similar computational costs.Because the VTN approximation does not need the area deriva-tives of the tesserae at the boundaries between spheres, it may beapplied to other cavity construction and tessellation procedures(instead of the GEPOL considered here) for gradient calculations.

The energy and gradients fluctuate during PCM geometry op-timizations (Figs. 1 and 3) due to the discrete change in thenumber, shapes, and areas of the boundary elements (tesserae)used in the approximate solution of the electrostatic Poisson equa-

tion. This can lead to severe practical problems such as slowconvergence, no convergence, and convergence to higher energeticconformations in geometry optimizations. The GEPOL-RT tessel-lation procedure, which uses high density of tesserae at boundariesbetween interlocking spheres, can lead to convergence for mostlarge biomolecules (Table 4). However, the computational cost isvery high due to the use of large numbers of tesserae. Furthermore,GEPOL-RT cannot handle the separation or merging of spheresthat causes especially large energy and gradient fluctuations. TheGEPOL-AS procedure, by scaling the areas of the tesserae that aretoo close to each other, can lead to convergence for all the

Figure 2. Model molecules extracted from X-ray crystal structure (PDB entry 1PPF) for Asp7, Glu10,Glu19, Asp27, and Glu43 of protein turkey ovumucoid third domain (OMTKY3). Only the protonated(acid) forms are shown. The positions of the atoms in bold are optimized at C-PCM/RHF/6-31G(d) levelof theory. See ref. 46 for further details about these models.


biomolecules considered (Table 4). Moreover, GEPOL-AS canhandle the separation or merging of spheres without additionalspheres. The combined use of GEPOL-AS and VTN gradients arerecommended for IEF-PCM/C-PCM geometry optimizations oflarge and flexible molecules.

Although satisfactory results can be obtained using the GE-POL-AS tessellation procedure, energy and gradient fluctuationsstill occur during PCM geometry optimization. This is the majorreason that the number of steps to reach the optimization toleranceis relatively large in some cases. Future studies will focus on bettertessellation procedures that generate even smoother potential en-ergy surfaces for molecules treated with PCMs.

Acknowledgments

This work was supported by a Research Innovation Award fromthe Research Corporation and a type G starter grant from thePetroleum Research Fund. H.L. gratefully acknowledges a predoc-toral fellowship from the Center for Biocatalysis and Bioprocess-ing at the University of Iowa. The calculations were performed onIBM RS/6000 workstations and an eight-node QuantumStationobtained through NSF, and on computers at the Advanced Bio-medical Computing Center (ABCC), SAIC-Frederick, National

Cancer Institute at Frederick, MD. The authors thank MichaelSchmidt for a careful reading of the manuscript.

Appendix

The following IEF-PCM equations can be found in Appendix A ofa previous work (ref. 42):

�A2

� D��1�� 1

� � 1

A2

� D�A�1SA�1q � �V (31)

q � q1 � q2 (32)

A�1SA�1q1 � �V (33)

�� 1

� � 1

I2

� DTA�1�q2 � �I

� � 1q1 (34)

Rearranging eq. (34) gives,

Figure 3. (a) Energy profiles and (b) rms gradient profiles for C-PCM geometry optimization of acid formAsp7 with various tessellations; (c) Energy profiles and (d) rms gradient profiles for C-PCM geometryoptimization of base from Glu43 with various tessellations. See Figure 2 for sketches of the molecules.


q1 � �� 1�� 1

� � 1

I2

� DTA�1�q2 � �� 1��I2

� DTA�1

�I

� � 1�q2 � �� 1��I2

� DTA�1�q2 � q2 (35)

Insert eq. (35) into (32):

q � �� 1��I2

� DTA�1�q2

� �� 1��A2

� D�T

A�1q2 (36)

Based on eq. (20), we have:

qTCxq � qT�A2

� D��1

Dx�A2

� D��1�� 1

� � 1

A2

� D�A�1SA�1q

� qT�A2

� D��1

DxA�1SA�1q � qT�A2

� D��1

� �� 1

� � 1

A2

� D�A�1SxA�1q (37)

Using eqs. (31), (32), and (38)

�� 1

� � 1

A2

� D� �A2

� D �A

� � 1(38)

to simplify or rearrange the first, second, and third terms of (37),respectively:

qTCxq � qT�A2

� D��1

Dx��V� � qT�A2

� D��1

DxA�1SA�1q1

� qT�A2

� D��1

DxA�1SA�1q2 � qTA�1SxA�1q

� qT�A2

� D��1� A� � 1�A�1SxA�1q (39)

According to eq. (33), the first term and the second term of (39) areidentical, thus canceling each other. Using eq. (36), the third termof (39) can be simplified:

�qT�A2

� D��1

DxA�1SA�1q2

� ��qT�A2

� D��1

DxA�1SA�1q2�TT

� ��DxA�1SA�1q2�T

� ��A2

� D��1�T

qT

� ��DxA�1SA�1q2�T��A

2� D��1�T

� �� 1��A2

� D�T

A�1q2�T

� �� 1�

� �DxA�1SA�1q2�TA�1q2�

T � �� 1�q2TA�1DxA�1SA�1q2 (40)

Similarly, the fifth term in (39) can also be simplified by using(36). Finally, we have:

qTCxq � �� 1�q2TA�1DxA�1SA�1q2

� qTA�1SxA�1q � q2TA�1SxA�1q (41)

Put (41) into (8), we have (21).

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improving the efficiency and convergence of geometry optimization with the polarizable continuum...

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