Generalized Molecular Mechanics Including Quantum Electronic

Chapter 11

Downloaded by UNIV OF MISSOURI COLUMBIA on October 29, 2017 | http://pubs.acs.org Publication Date: December 28, 1998 | doi: 10.1021/bk-1998-0712.ch011

Generalized Molecular Mechanics Including Quantum Electronic Structure Variation of Polar Solvents: An Overview 1

1

1

Hyung J. Kim , Badry D. Bursulaya , Jonggu Jeon , and Dominic A. Zichi

2

1

Department of Chemistry, Carnegie Mellon University, 4400 Fifth Avenue, Pittsburgh, PA 15213-2683 NeXstar Pharmaceuticals, Inc., 2860 Wilderness Place, Boulder, CO 80301

2

A brief account is given of recent theoretical work on the computationally-efficient quantum mechanical description of the solvent electronic structure variation via a truncated adiabatic basis-set representation. B y the inclusion of both linear and nonlinear solvent electronic response, this goes beyond many existing classically polarizable solvent descriptions, widely used in simulation studies. Its implementation with Molecular Dynamics (MD) simulation techniques and application to liquid water are described.

Due to its essential role in condensed-phase processes, solvation has received extensive theoretical and experimental attention for quite some time. Among others, the computer simulation methods have been very instrumental in revealing molecular-level details of solvation (1 -3). In recent years, considerable theoretical efforts have been focused on the construction of electronically polarizable solvent descriptions (4 -23). With a few exceptions (19,22,23), classical electrostatics are employed for polarizability, so that the induced dipole moment for each molecule is proportional to the local electric field arising from its surroundings. While this accounts for the induction effects to the leading order in the electric field, the higher-order effects are not included. Another difficulty is the electronic transitions and accompanying relaxation. Even though ab initio M D couched in density functional theory provides significant improvement in the solvent electronic description (24 ), intensive C P U demand restricts its application to the simulations of small systems for a short period of time (25 -30). Also, its generic difficulty with the excited states does not allow for electronic transitions, relevant for various electronic spectroscopies. Therefore it seems worthwhile to develop a theoretical description that captures important electronic features in a quantum mechanical way and yet is computationally efficient to allow for extended simulation studies of macroscopic systems. 172

©1998 American Chemical Society Gao and Thompson; Combined Quantum Mechanical and Molecular Mechanical Methods ACS Symposium Series; American Chemical Society: Washington, DC, 1998.

173 In this contribution, we give a brief account of our recent efforts to address the above-mentioned solvent electronic aspects in the context of M D simulations. B y employing a truncated adiabatic basis-set (TAB) description, we incorporate both the linear and nonlinear solvent electronic response into our theory. This is applied to liquid water to study its solvation properties via M D . Here we give only enough description to make our accounts reasonably self-contained; for further details, the reader is referred to the original papers (31 -33).


Theoretical Formulation In the T A B formulation, the electronic wave function of each solvent molecule is represented as a linear combination of a few vacuum energy eigenstates

# = Σ ^ = --^4.(2) j i(>j) i j i(>j) I* î\ where p* is the electric dipole operator for i , r» is its position vector, i/Jj and Tij are, respectively, the L J interaction and dipole tensor between i and j, ή 7

r

Gao and Thompson; Combined Quantum Mechanical and Molecular Mechanical Methods ACS Symposium Series; American Chemical Society: Washington, DC, 1998.

174 is a unit vector in the direction of r» - fj and J is a unit matrix. In the T A B representation consisting of M basis functions, h* is a diagonal MxM matrix. By contrast, p is, in general, not diagonal; its off-diagonal elements correspond to the transition dipole moments between different 0^'s. For clarity, we add a remark here: in contrast to repulsion arising from electron exchange, the dispersion can be, to a large degree, accounted for in the T A B description by including the intermolecular electronic correlation effects via, e.g., the M P 2 method (31 ). However, in this initial study with low-level electronic structure calculations, we take into account the dispersion through the L J potentials instead. For simplicity, we further assume that υ£ do not depend on the electronic states. We determine the system electronic wave function and energy eigenvalue in the self-consistent field (SCF) approximation


1

3

Ν

I *sc

> = Π 11>*

>

;

^el

I * s c > = Esc I * s c > ;

»=1

U = Esc + Σ Σ «La > J i{>j)

(3) where Ε$ and U are, respectively, the ground-state S C F and total potential energies of the system. This involves solving a set of "Fock" equations α

/*|^>=[Λ' +

οί ]|^>= '|^>; Μ

v\ =

€

M

Σ

J

-Tjrp\

(4)

i(^)

where v\ is an effective 1M operator representing the electrostatic interaction between molecule i and the self-consistent field arising from all other molecules. The structure of equation 4 is very similar to that of the Hartree-Fock quantum chemistry method (41)· Except for the absence of exchange terms in equation 4, these two theories are nearly isomorphic to each other (molecule electron and T A B +-» electronic basis set). We arrange the 1M SCF solutions for each molecule in the order of increasing energy and denote their states and energies as | a > and e (a = 0 , 1 , . . . , M — 1). Then the ground electronic state | \I>sc > for the entire solvent system in the SCF approximation results when every molecule is in its ground state 10* > M

1

l

a

Ν

I *sc

>= Π1

0
;

(5)

/ * l *sc > = 4 I ^sc > ;

Eg. = < Φ

;

8 0

\Ê \ *sc > , el

i where E is the sum of the 1M ground state energies eô's. The ground-state dipole moment μ and electronic polarizability tensor a of molecule i in solution are given by (31 ) G

1

i

i

i

/x = < 0 | p i 0 >

;

π

< = 2 ^ ^ 0 ; e

a(^0) a

a* = π* + 5 > * - Z V ni.Tfi-n* J

C

= < 0'

1

a* > ;

0

Σ π < " τ « " ^ 'Τ^π^Τ^π' j

+ ... ,

Μ


(6)

175 where the terms involving arise from the interaction-induced effects; i.e., a change in μ induces a dipole change for j through electrostatic interactions. Thus π* is the polarizability of i that would result if the interaction-induced dependence of the electronic structure of the surrounding molecules is neglected. Even though it is a 1M quantity, π* depends on the intermolecular interactions through e and $Jo. Thus, π* generally differs from the gas-phase polarizability and so does a*. Also, both π* and a* vary with i because different molecules are subject to differing local solvation environments. This is closely related to nonlinear hyperpolarizability, absent in many classically polarizable potential models. The total electronic polarizability tensor 17 of the solvent is given by 1

l


a

Π = Σ

π

Ref. 58. The value in parentheses is determined with the Clausius-Mossotti equation 12. ) Ref. 61 . > Ref. 60. 8

3

2

1

tr

b

c)

d

e

We can evaluate the optical dielectric constant via (62) 6oo - 1

1 4π _

_

ΑπΝ _

,

s

λ

where we have again neglected the interaction-induced effects in passage to the last expression. Here V is the volume of the simulation cell. With equation 11, T A B / 1 0 yields £oo — 1-56. If we instead employ the Clausius-Mossotti equation e « - l

4xAT_

S )

(

1

2

)

we obtain ε » = 1.69. While this is in fair agreement with the experimental value ε » = 1.79 (61 ), the T A B description underestimates e » due to the truncation


181 of the excited states. This state of affairs could be, in principle, improved by increasing the number of T A B functions. The static dielectric constant ε , on the other hand, can be determined in the linear response regime via (63 -65) 0


e

o = e~ + 3 ^ ( < M

2 >

e

, - < M > y

;

Μ= Σ μ \

(13)

where ks is Boltzmann's constant, Τ is the temperature and M is the total dipole moment of water. We found £o = 7 9 ± 10 for TAB/10, which is in excellent accord with the experimental result. A b s o r p t i o n S p e c t r u m . We now proceed to the distribution of excitation en ergies to gain insight into absorption spectroscopy. In this initial attempt, we neglect the electronic relaxation effects and consider only the 1M S C F excited states. [The electronic relaxation could be partially accounted for via the 1MCI method in equation 8.] The results are displayed in Figure 3. The first peak near 8.2 eV shows reasonable agreement with the absorption band of liquid water around 8.2-8.5 eV (66 -70). Also the broad character of the band, arising from the inhomogeneous distribution of differing local solvent environments, is well described by the T A B / 1 0 model. Compared with the lowest electronic transition Χ Ai —> A Bi with energy 7.6 eV in vacuum [cf. Table IIA], the corresponding transition in solution is blue-shifted by ~ 0.6 eV, consonant with experiments. While the inclusion of electronic relaxation may shift the absorption band to a lower energy compared to that in Figure 3, the qualitative and even semiquan titative features observed here will remain essentially unchanged because of the nonequilibrium solvation effects arising from the solvent permanent dipoles (71). To the best of our knowledge, this is the first simulation study that yields semi quantitative agreement with photoabsorption measurements [cf. refs. 19 and 26 ]. l

l

0.8

6

8

10

12

14

energy gap (eV) Figure 3: The photoabsorption spectrum of T A B / 1 0 water with the neglect of the electronic relaxation effects.


182 O p t i c a l K e r r Effect Spectroscopy. Finally, we consider optical Kerr effect (OKE) spectroscopy, characterized by third-order nuclear response (72 ) {3)

R (t) = -j^^

< U * W IWO) >

(t > ο ) ,

(14)

where (ab = xy, xz, yz) is an off-diagonal component of Π in the lab frame. The M D result for the T A B / 1 0 Kerr response function, Also presented there is a model response function, R^ \ obtained with fixed nonfluctuating electronic po larizability. was evaluated by replacing fluctuating π* in equation 7 with its solution-phase ensemble average, fr [equation 10]; this will be referred to as the nonfluctuating (NF) model. To separate out the polarizability effects on dynam ics, we used the T A B / 1 0 trajectory to compute Λ ; thus the underlying nuclear dynamics for both the T A B / 1 0 and N F models are exactly the same. Therefore, the only difference between the two is that the polarizability of T A B / 1 0 fluc tuates dynamically with the solvent configuration, whereas that of N F remains unchanged. As noted above, their anisotropy is very close to Murphy's measured value (55). Thus T A B / 1 0 and N F are nearly isotropically polarizable in solution.


3

s

( 3 )

ο '»•••»••••' 0 0.05 0.1

0.15

, ι , , • .ι 0.2 0.25

t (fs) Figure 4: The third-order nuclear Kerr response function R^(t) [equation 14] for T A B / 1 0 water (—). For comparison, the Kerr response R^(t) of the N F model with fixed electronic polarizability π is also shown (·••)· β

One of the most pronounced features in Figure 4 is that Kerr response of the T A B / 1 0 and N F models differs markedly. After the initial peak around 15 fs, the latter yields no distinctive structures in R^ up to ~ 150 fs; this is in striking contrast with the R^ peaks around 50 and 95 fs. Also the short-time behavior of N F is considerably slower than that of TAB/10. The reader should note that this dramatic difference in Kerr response arises solely from the polarizability fluctuations (or lack thereof). As shown in ref. 33, this is due to the instantaneous adjustment of the TAB/10 molecular polarizability to the fluctuating local electric



183 field. This nonlinear solvent electronic response enhances the librational character of, and introduces the short-time oscillatory structures into, Kerr response of liquid water (33 ). We also point out that the T A B / 1 0 response function compares well with recent O K E measurements (73 -76). The structures around 20 and 200 fs determined from experiments are in reasonable accord with those of RW at ~ 12 and ~ 150 fs. Also, the existence of a peak near 50 fs in the latter is in good agreement with the experimental findings (74 -76). Therefore, despite its nearly isotropic polarizability in solution, the T A B / 1 0 potential reproduces the experimental results on nuclear Kerr response fairly well due to the inclusion of polarizability fluctuations. For a more detailed description, the reader is referred to ref. 33 . [For previous M D studies on nonlinear electronic spectroscopy on water, see, for example, refs. 77 -80.] Concluding Remarks In this article, we outlined our recent efforts to include the solvent electronic polarizability in M D simulations. By employing an efficient quantum mechanical description in a truncated adiabatic basis-set representation, both linear and nonlinear aspects of the solvent electronic structure variation were accounted for. Its application to liquid water was effected via the T A B / 1 0 potential model with 10 basis functions. The parametrization of the basis functions was couched in terms of the C A S S C F ab initio results for an isolated water molecule with the aid of experimental information. In the simulations, the diagonal and overlap charge distributions of the T A B functions were represented by partial charges on the five interaction sites of water; two of them are fictitious sites located off the molecular plane to describe out-of-plane polarizability. The ground electronic wave function and energy for the entire solvent system were determined at the SCF level and the intermolecular forces were evaluated using the Hellmann-Feynman theorem. It was found that the T A B / 1 0 potential provides a good overall descrip tion for bulk water. Its predictions for structural, dynamic, spectroscopic and dielectric properties of water are in good agreement with experiments. Its translational diffusion coefficient and static dielectric constant were found to be, re spectively, D « 2.4 Fick and £o « 79; the corresponding experimental results are D = 2.3 Fick and ε = 78. The first absorption band centered around 8.2 eV and associated blue-shift compare well with the experimental findings. Also, despite its small polarizability anisotropy, T A B / 1 0 water correctly cap tures experimentally-observed short-time behavior of nuclear Kerr response—in particular, the structure around 50-60 fs; this is usually not reproduced with classically polarizable solvent descriptions. Finally, compared with a nonpolarizable model with 5 interaction sites, the performance of T A B / 1 0 at the S C F level was found to be slow by a factor of ~ 8. This is mainly due to the repeated diagonalizations of the Fock matrices [equation 4] to determine the solvent electronic structure. One promising aspect, though, is that our generalized molecular mechanics algorithm would allow a time step larger than 2 fs since the electronic degrees of freedom are optimized accurately by solving the Fock equations. Also in view of its various advantages, tr

tr

0


184 e.g., nonlinear electronic response and spectroscopy, we believe that the T A B formulation provides a promising and viable alternative to classical polarizable solvent descriptions and density functional M D . Acknowledgments This work was supported in part by NSF Grant No. CHE-9412035.


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Generalized Molecular Mechanics Including Quantum Electronic

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