Dielectric solvation dynamics of molecules of arbitrary shape and charge distribution

Transcription

1 JOURNAL OF CHEMICAL PHYIC VOLUME 08, NUMBER 6 8 FEBRUARY 998 Dielectric solvation dynamics of molecules of arbitrary shape and charge distribution Xueyu ong and David Chandler Department of Chemistry, University of California, Berkeley, California 9470 Received 5 eptember 997; accepted 3 November 997 A new perspective of dielectric contuum theory is discussed. From this perspective a dynamical generalization of a boundary element algorithm is derived. This generalization is applied to compute the solvation dynamics relaxation function for chromophores various solvents. Employg quantum chemical estimates of the chromophore s charge distribution, the Richards Lee estimate of its van der Waals surface, and the measured frequency dependent dielectric constant of the pure solvent, the calculated relaxation functions agree closely with those determed by experiments. 998 American Institute of Physics I. INTRODUCTION This paper describes a generally applicable algorithm for applyg dielectric contuum theory and its generalizations to solvation and solvation dynamics with realistic models of solute molecular shape and charge distribution. The algorithm is based on the Gaussian field formulation of traditional dielectric contuum theory. For static applications, our method is closely related to the widely used boundary element method for electrostatic calculations. 5 But our method offers a dynamical generalization and possible extension to molecular models of a solvent. As an application, we give a comparison with recent experimental measurements of solvation relaxation functions for coumar343 C343 ) water 6 and coumar53 C53 methanol and acetonitrile. 7 The puts to our calculations are the charge distribution change of the chromophore from the electronic ground state to its first electronically excited state, the chromophore s molecular surface, 8 and the frequency dependent dielectric constant of the solvent. The first of these is estimated from quantum chemistry calculations; the second is determed from standard atomic van der Waals radii; the third is found from experiments. With these gredients, good agreement between experiments and theory is found. Other standard approximations e.g., the dipole a sphere model, and the uniform dielectric approximation 9 are shown to be less satisfactory. We beg with a general relation for the response function of a solvent the presence of a solute, Eq. below. We derive the connection between this relationship and the boundary element method. 3 5 In ec. III, we test our methodology for the case of a charge distribution a spherical dielectric cavity. Kirkwood s analytical result for this case is reproduced. In ec. IV, we apply our formulation to several experimental systems where the solute is defitely not spherical. The paper is concluded ec. V. II. INTEGRAL EQUATION FORMULATION In Ref., the followg general relationship is derived m r,r;sr,r;s drdrr,r;s r,r;s r,r;s. Here, (m) (r,r;s) is the solvent dipole density response function of the solution, and (r,r;s) is that same quantity, but for the pure solvent. The dipole density is a vector. Hence, (m) (r,r;s) and (r,r;s) are 33 matrices. The labelg the tegration symbol means the tegration is limited to the region occupied by the solute and from which the solvent dipole density is expelled. The verse of the matrix, (r,r;s), is defed as dr r,r;s r,r;srri, where I is the 33 identity matrix associated with the Cartesian coordates of a three-dimensional system, and (r,r;s) is(r,r;s) when both r and r are with the region, and it is zero otherwise. In dielectric contuum theory, the response function is,0 r,r;s 4s s srri s 4 rr, where (s) is dielectric constant as a function of frequency, s. Assumg /98/08(6)/594/7/$ American Institute of Physics

2 J. Chem. Phys., Vol. 08, No. 6, 8 February 998 X. ong and D. Chandler 595 r,r;s s 4s s rri s 4s rr fr,r;s, we can fd an equation for f (r,r;s) after adoptg a specific form for (r,r;s). By usg the dielectric contuum form, Eq. 3, Eq. gives sfr,r;s s 4s s 4 s 4 rr s dr rr rr dr f r,r;s rr 0. This result is an tegral equation for the short ranged function f (r,r;s). To simplify this last equation, the volume tegration can be converted to a surface tegral. In particular, if r is with the region, and if rr 0 is on the surface,, surroundg the region, then f (r,r 0 ;s) satisfies s f r,r 0 ;s s 4 da R fr,r;s n RR 0 s 4 da s R RR 0 n Rr 0. Here, da R is the normal area element at position R of the surface and n stands for the outward normal gradient at R. This equation is derived usg Green s first identity. Once f (r,r;s) is solved from above equation, the short ranged f (r,r;s) for both r and r with the region is obtaed from f r,r;s s 4 da R fr,r n s 4 Rr da s R Rr n Rr. 7 The calculation of (r,r;s) is thus reduced to a twodimensional tegral equation, Eq. 6. In general, an analytical solution to Eq. 6 is difficult to fd. Except for some special geometries, such as a spherical cavity, we need to solve the equation numerically. One numerical method of solution is discussed ec. IV. Once we solve Eq. 6, the response function of the system can be calculated from Eqs. 4, 7 and and all the properties of the system can be calculated from it through lear response theory. For example, consider a charge distribution created at time t0 side a dielectric cavity. The Fourier transform of the time dependent solvation energy of the charge distribution due to the dielectric is Es drdr s r i,r j rr i m r,r;s q j rr j, 8 where r i is the position of the ith charge, q i. As shown Appendix A, this equation for E(s) can be reduced through the application of Eqs. and 4, givg Es 4s q s s i q j f r i,r j ;s. 9 r i,r j This equation is one of our major results. It relates the calculation of the time dependent non-equilibrium solvation energy to that of a short ranged non-sgular function. With Eqs. 6 and 7, the result is a time dependent generalization of the boundary element method for performg dielectric contuum calculations. This connection can be seen by considerg r,r;s n f r,r;s s 4s n rr. 0 In Appendix B, it is shown that (r,r;s) satisfies the followg tegral equation: r,r 0 ;s s da s R r,r;s n RR 0 s s n 0 rr 0 0, where n0 denotes the outward normal gradient at R 0 and the solvation energy can be expressed terms of (r,r;s), Es q s r i,r j j da R q i r i,r;s Rr j. When comparg with the boundary element method, 3 5 we can identify (r,r;s) as the duced charge at R. Equation 9 Ref. 3b is equivalent to our Eq.. In general, the formulation of this section is not confed to the contuum theory of a non-ionic dielectric. For example, place of Eq. 3, one may use a response function matrix appropriate to a homogeneous ionic solution. A short length scale with molecular detail could also be cluded. Equation 4 could be written, but the resultg tegral equation for f (r,r;s) will be modified due to the features (r,r;s) referrg to ionic strength and molecular detail. For the applications given this paper, however, we confe ourselves to the case of a non-ionic dielectric contuum solvent. III. PHERICAL CAVITY For a spherical cavity, Eq. 6 can be solved analytically. On expandg the f function spherical harmonics Y lm (,) for a spherical cavity with radius R, q i

3 596 J. Chem. Phys., Vol. 08, No. 6, 8 February 998 X. ong and D. Chandler f r,r;s lm B l r,r;sy lm *,Y lm,, 3 where r, and are spherical coordates of r. ubstitution to Eq. 6 yields B l r,r;s s l s l lsl Combg Eq. 4 with Eq. 7 gives B l r,r;s s s r l. 4 l R l rr l l lsl R. l 5 We use this result to calculate solvation energy of a charge distribution side a spherical dielectric cavity. In particular, from Eq. 9, the solvation energy for a unit charge at position b this cavity is Es sr s l lslr b l. 6 This equation is precisely the Kirkwood result obtaed by solvg the Poisson equation, but generalized to the frequency dependent case. The exact solvation energy, Eq. 6, can be compared with that obtaed from the uniform dielectric approximation UDA. In the UDA, the solvation energy of a unit charge at position b a dielectric cavity is E UDA s s out drdr rb r,r;s rb, 7 where the tegration limit out dicates the tegration over r,r outside the solute dielectric cavity. For a spherical cavity, the tegration can be done analytically, givg E UDA s sr l s llls R b l. 8 l From these expressions, the solvation relaxation function, 6 FIG.. olvation relaxation function, (t), calculated from the exact result from Eq. 6, dashed le and the UDA from Eq. 8, solid le. The frequency dependent Debye dielectric constant, (s) ( 0 )/( is D ) with 4. and , is used the calculation. Time t is scaled by the longitudal relaxation time, L D ( )/( 0 ). Panel a is for a charge located at 0.5 Å from the center of a sphercal cavity with 0 Å radius. Panel b is for a charge beg at 7.5 Å from the orig. dielectric cavity. To demonstrate our algorithm we first calculate the solvation energy of a charge distribution side a spherical cavity, where the analytical solution is known Eq. 6. Then the numerical algorithm is applied to three ex- t ẼtẼ, 9 Ẽ0Ẽ can be calculated. Here, Ẽ(t) is the verse Fourier transform of E(s) at time t. The results of the calculation are shown Fig.. It is seen that UDA is serious error if the charge is near the dielectric cavity surface. The exact results show that a general distribution of charges can duce a range of relaxation times. IV. NUMERICAL OLUTION AND COMPARION WITH EXPERIMENT We now discuss how to solve the tegral Eq. numerically and combe it with Eq. to calculate the time dependent solvation energy of a charge distribution side a FIG.. The convergence test of our numerical algorithm for the solvation energy as a function of number of triangulations. The test system has the followg charge distribution: a unit charge at 8.0,0.0,0.0, a two unit charge at 8.0,0.0,0.0 and a negtive two unit charge at 8.0,.0,0.0, all side a spherical cavity of a radius 0 Å centered at the orig. The dielectric constant of the outside dielectric i.e., the solvent is 80. The le is the exact result from Eq. 6 and the data pots are from numerical calculation.

4 J. Chem. Phys., Vol. 08, No. 6, 8 February 998 X. ong and D. Chandler 597 TABLE I. The atomic charge distribution of C343. a erial No. Atom Type Coordates of Atoms Ground state charge q b C C C C C C C C C O N C H H C O C O O C C C C H H H H H H H H H H H H a The charge distributions given this table, based upon the procedures noted the text, were provided by Mark Maroncelli. imilar formation for C53 is given Ref. 7. b q the charge difference, units of the charge of an electron, between ground state and excited state. perimental systems. The solvation relaxation function is calculated from the time dependent solvation energy and a direct comparison with experimental results of C343 water, 6 C53 methanol and C53 acetonitrile 7 is made. The numerical method to solve the tegral Eq. over a surface is based on collocation methods by Atkson and co-workers. 3 Basis functions prescribed Ref. 3 are set up over the surface by triangulation, and the tegral equation is converted to a system of lear equations. In Fig., a convergence test is shown to dicate that good agreement with the analytical result can be obtaed with a reasonable number of triangulations over the surface, even for severe situations e.g., 80, and charges located side but about Å away from the surface of a 0 Å spherical dielectric cavity. For our solvation dynamics applications, the time dependent solvation energy is obtaed by the followg procedure: For a molecule like C343, we use the Lee and Richards molecular surface. 8 Triangulation over this surface is done by Zauhar s method. 4 The probe radius of water is.4 Å and the van der Waals radii of the atoms are taken from the CHARMM parameter package. 5 The atomic charge distribution and its change due to electronic excitation of C343 are obtaed from electrostatic fits to a semi-empirical modified neglect of differential overlap MNDO wave function 6, calculated usg the AMPAC program package. The resultg partial charges are given the entrees to Table I. For C53, a similarly derived set of partial charges are tabulated Ref. 7. The frequency dependent dielectric constant of solvents, (s), are taken from Refs. 9 and 0. With the above formation, the time dependent solvation energy has been calculated by carryg out the verse Fourier transform of Eq.. The results are shown Figs. 3, 4 and 5. It is clear that all of the major features experimental results, the itial ertial decay and the multiple exponential long time relaxation, are reproduced almost quantitatively by our calculations. Also shown Figs. 3, 4 and 5 are the results where a dipole with a spherical cavity is used as a model of the solute. 8 In this simple model, (t) is the verse Fourier transform of ((s))/((s)). It is dependent of the dipole magnitude and dependent of the cavity size. The comparison between this simple model and our detailed calculations shows that (t) can be relatively sensitive to the detailed charge distribution and the shape of the probe solute. This sensitivity does not necessarily

5 598 J. Chem. Phys., Vol. 08, No. 6, 8 February 998 X. ong and D. Chandler FIG. 3. The comparison of solvation relaxation function between theory and experiments for C343 water. The short dash curve is from experiments Ref. 6, the long dash curve is from our calculation, and the solid curve is from a dipole a spherical cavity model Ref. 8. The dielectric data are from Ref. 9. FIG. 5. The comparison of solvation relaxation function between theory and experiments for C53 methanol. The short dash curve is from experiments Ref. 7, the long dash curve is from our calculation, and the solid curve is from a dipole a spherical cavity model. The dielectric data are from Ref. 0. hold for the absolute size of the solvation energy. For example, the charge distribution change for C343 is about.5 D. To reproduce the value of E(0) with this dipole a spherical cavity, the effective cavity radius would be about 3.7 Å. This radius size seems unphysically small view of the space fillg model of C343 see Fig. 6. Equation is lear (r,r;s). Therefore, this one equation is to be solved whether there are one, two, or any number of charges. ce solvg this equation represents the domant time consumg step implementg our approach the solvation energy calculation is almost dependent of the numbers of charges side a dielectric cavity. As the size of the solute creases, the number of triangulations needed to obta an accurate result creases. For large solutes, this feature can be a major disadvantage of boundary element methods. ome new algorithms have been designed to overcome this disadvantage. 5 One may use the duced dipole on the dielectric cavity surface to account for the effect of dielectric. We show Appendix B that the duced dipole method is equivalent to the duced charge scheme. In numerical tests for charge distributions a spherical cavity, we fd that these two methods give the same result with equal triangulations and the computational time is almost the same. Thus, the duced dipole scheme offers no obvious advantages. V. CONCLUDING REMARK In this report we have explored the utility of the Gaussian field theoretic formulation of dielectric contuum theory. With this formulation, we have derived a practical algorithm to calculate time dependent solvation energy for an arbitrary charge distribution a dielectric cavity with realistic molecular shape. While it has much common with the well known boundary element method, the formulation generalizes the method so as to treat solvation dynamics. Further, as discussed ec. II, it seems possible to extend the method so as to treat the effects of short length scale solvent structure and non-zero ionic strength. The response of a Gaussian field model is by defition lear. This feature may be a significant limitation of our approach when non-lear response becomes important. It may be possible to describe the effects of non learities FIG. 4. The comparison of solvation relaxation function between theory and experiments for C53 acetonitrile. The short dash curve is from experiments Ref. 7, the long dash curve is from our calculation, and the solid curve is from a dipole a spherical cavity model. The dielectric data are from Ref. 0. FIG. 6. A space-fillg model of C343. The dashed le shows the size of the sphere where the dipole a spherical cavity model gives the same solvation energy as the full calculation.

6 J. Chem. Phys., Vol. 08, No. 6, 8 February 998 X. ong and D. Chandler 599 through perturbation theory, where a general Gaussian field model serves as a reference. This possibility merits consideration the future. ACKNOWLEDGMENT This research has been supported by the Basic Energy cience Division of the U Department of Energy. We are debted to Randy Zauhar for providg us with the triangulation computer program package for determg molecular surfaces. The charge distribution of C343 and C53 used the paper are kdly provided by Mark Maroncelli. APPENDIX: DERIVATION OF EQ. 9 We focus on a situation where there is one charge a dielectric cavity. The generalization to arbitrary charge distribution is straightforward. Consider a unit charge at position a side a dielectric cavity. Beg with Eq. and defe EE 0 E c, where E 0 is the contribution from the first term of (m) and E c comes from the contribution of the second term of (m). Usg Eq. 3 for, wefd E 0 s sdrdr ra r,r;s ra s se s, A where E s is the self-energy of a unit charge. imilarly, E c s sdrdr ra drdrrr;s r,r;s rr;s ra. A By usg Eq. 3 for, the tegration over r,r can be performed, givg E c s s s drdr 4s ra r,r;s ra. A3 With Eq. 4, the tegration over r,r can be done, to yield E c s s s E s s 4s s f a,a;s. A4 Combg Eqs. A and A4, the self-energy contributions cancel, givg Es s 4s s fa,a;s. A5 APPENDIX B: DERIVATION OF EQ. AND From Eq. 5 we can easily show that f (r,r;s) satisfies f r,r;s0. B Thus, by Green s theorem we fd da R fr,r;s n RR 0 da R n fr,r;s RR 0 fr,r 0 ;s, B da R fr,r;s n Rr da R n fr,r;s Rr 4fr,r;s. B3 Combg Eqs. B, 6 and 0 with the followg identity n0 da R Ir,R;s RR 0 da R Ir,R;s n RR 0 Ir,R 0;s, B4 where I(r,R;s) is an arbitrary function of r,r and s, then, Eq. can be obtaed. Equation can be derived from Eqs. 7, 9, 0 and B3. Consider pr,r;s f r,r;s s 4s rr. B5 With Eq. 6, it can be shown that p(r,r;s) satisfies the followg tegral equation: pr,r 0 ;s s da s R pr,r;s n RR 0 s s rr 0 0, B6 and the solvation energy can be expressed terms of p(r,r;s), q j da R q i pr i,r;s n Rr j. B7 Es s r i,r j It is thus clear that p(r,r;s) is the duced dipole the normal direction of R due to a unit charge at position r side the dielectric cavity sce the duced charge, (r,r;s), is related to the duced dipole p(r,r;s) the normal direction of R by (r,r;s) n p(r,r;s). X. ong, D. Chandler, and R. A. Marcus, J. Phys. Chem. 00, J. Tomasi and M. Persico, Chem. Rev. 94, a R. J. Zauhar and R.. Morgan, J. Mol. Biol. 86, ; b R. J. Zauhar and R.. Morgan, J. Comput. Chem. 9, B. J. Yoon and A. M. Lenhoff, J. Comput. Chem., ; M.E. Davis and J. A. McCammon, Chem. Rev. 90, ; R. Bharadwaj,

7 600 J. Chem. Phys., Vol. 08, No. 6, 8 February 998 X. ong and D. Chandler A. Wdemuth,. ridharan, B. Honig, and A. Nicholls, J. Comput. Chem. 6, L. R. Pratt, G. J. Tawa, G. Hummer, A. E. Garcia, and. A. Corcelli, Int. J. Quantum Chem. 64, 997, and references there. 6 R. Jimenez, G. R. Flemg, P. V. Kumar, and M. Maroncelli, Nature London 369, M. L. Horng, J. A. Gardecki, A. Papazyan, and M. Maroncelli, J. Phys. Chem. 99, F. M. Richards, Annu. Rev. Biophys. Bioeng. 6, 5 977; M. L. Connolly, J. Appl. Crystallogr. 6, B. Bagchi and A. Chandra, J. Chem. Phys. 90, ; L. E. Fried and. Mukamel, ibid. 93, ; F. O. Raeri, Y. Zhou, H. L. Friedman, and G. tell, Chem. Phys. 5, M.. Wertheim, Annu. Rev. Phys. Chem. 30, J. D. Jackson, Classical Electrodynamics, nd ed. Wiley, New York, 975. J. G. Kirkwood, J. Chem. Phys., K. Atkson, IAM oc. Ind. Appl. Math. J. ci. tat. Comput. 5, R. J. Zauhar, J. Comp-Aided Mol. Des. 9, ; R. J. Zauhar and A. Varnek, J. Comput. Chem. 7, B. R. Brooks, R. E. Bruccoleri, B. D. Olafson, D. J. tates,. wamathan, and M. Karplus, J. Comput. Chem. 4, F. A. Momany, J. Phys. Chem. 8, P. V. Kumar and M. Maroncelli, J. Chem. Phys. 03, C. Hsu, X. ong, and R. A. Marcus, J. Phys. Chem. 0, X. ong and R. A. Marcus, J. Chem. Phys. 99, C. Hsu and R. A. Marcus unpublished. T. Fonseca and B. M. Landanyi, J. Phys. Chem. 95, 6 99; M.. kaf and B. M. Landanyi, ibid. 00, ; M. Maroncelli and G. R. Flemg, J. Chem. Phys. 89, ; M. Maroncelli, ibid. 94, M. A. Jaswon and G. T. ymm, Integral Equation Methods Potential Theory and Electrostatics Academic, London, 977.