2736
J. Phys. Chem. 1992, 96, 2136-2142
An Excess ftectton-llkahanal F8eudopo;tentlal Maria Hikzer,* Witold M. Bartczak, and Miroslaw Sopek Institute of:Applied Radiation Chemistry, Technical University (Politechnika), -2, (Received: August 30, 1991)
Wr6blewskiego 15, Poland
A pseudopotential describing the interaction between an excess electron and a methanol molecule in the electronic ground state is presented. The potential consists of electrostatic, repulsion, exchange, and adiabatic polarization terms. The potential is suitable for the description of the excess electron states in a methanol matrix and can be used in quantum simulations of the electron solvation in methanol.
I. Introduction The strength of the electronsolvent interaction determines the properties of the solvated electron and, in particular, the spatial extension of bound excess-electronic states. The solvated electron (e;) has been the object of numerous theoretical studies. The history of the model calculations of the solvated electron began with a particle-in-box model, where the interactions between the electron and the solvent molecule are described in terms of a spherical box type potential.' The model emphasized the short-range interactions, similar to the more recent structural models which describe a cluster of molecules plus the excess electron by ab initio methqds of quantum chemistry.2d In contrast to the former class of e; models, the dielectric continuum models emphasize the long-range potentials which are experienced by the excess electr~n.'.~In these latter models, the electron environment is represented by a polarizable continuous dielectric characterized by its static and optical dielectricconstants. The trapping potential arises from the pdarization of the medium induced by the electron. Both the short- and long-range interactions have been included in the semicontinuum models9J0in which the trap-forming molecules have been treated on a microscopic level in the point-dipole apprgximation and the interactions with further molecules are defined similarly as in the continuum models. All these models, besides treating the electron-medium potential in a very approximate way, usually propose a single optimized structure of the electron trap. On the other hand, an adequate description of the electron solvated in amhrphous media should take into account molecular disorder of the medium, which is reflected in a statistical variety of the electron traps. The statistical approach to the 4- problem is based either on direct transformation of the distributions of random parameters in the e; Hamiltonian" or on the quantum simulation techn i q u e ~ . ' * - ' ~The ~ ~latter ~ calcuIations extend the classical Monte (1) Ogg, R. S. Phys. Rev. 1948, 69, 668. (2) Newton, M. J . Phys. Chem. 1975,79,2795; J . Chem. Phys. 1973,58, 5833. (3) Naleway, C. A.; Schwartz, M. E. J. Phys. Chem. 1972, 76, 3905. (4) Natori, M. J. Phys. SOC.Jpn. 1969, 27, 1309. (5) Noel, J. 0.; Morokuma, K. J . Phys. Chem. 1977, 81, 2295. (6) Rao, B. K.; Kestner, N. R. J . Chem. Phys. 1984, 80, 1587. (7) Jortner, J. J. Chem. Phys. 1959, 30, 839. (8) Fueki, K.; Feng, D.-F.; Kevan, L. J. Phys. Chem. 1970, 74, 1976. (9) Copeland, D. A.; Kestner, N. R.; Jortner, J. J . Chem. Phys. 1970,53, 1189. (10) Fueki, K.; Feng, D.-F.; Kevan, L. J. Am. Chem. Soc. 1973.95, 1398. (1 1) Bartczak, W. M.; Hilczer, M.; Kroh, J. J. Phys. Chem. 1987, 91, 3834. Hilczer, M.; Bartczak, W. M. J . Phys. Chem. 1990, 94, 6165. (12) Jonah, C. D.; Romero, C.; Rahman, A. Chem. Phys. Leu. 1986,123, 209. Romero, C.; Jonah, C. D. J. Chem. Phys. 1989, 90, 1877. (13) Sprik, M.; Impley, R. W.; Klein, M. L. J. Star. Phys. 1986,43,967. Berne, B. J.; Thirumalai, D. Annu. Rev. Phys. Chem. 1986, 37, 401 and
references cited therein. (14) (a) Schnitker, J.; Rossky, P. J. J. Chem. Phys. 1987.86, 3471. (b) Rossky, P. J.; Schnitker, J. J . Phys. Chem. 1988, 92, 4277. (c) Schnitker, J.; Motakabbir, K.; Rossky, P. J.; Friesner, R. A. Phys. Rev. Left. 1988,60, 456.
Carlo or molecular dynamics methods to include the interaction of an excess electron with the simulated matrix.19 Recent years have also brought progress in the theoretical studies of the fluctuations of potential experienced by an excess electron in liquids. These calculations produce the statistics of the microscopic trapping sites (the preexisting traps) by analysis of the data on the solvent molecular structure obtained by the computer simulation methods. The simulations have been performed either for small molecular clusters20 or for large matrices containing several hundreds of molecules.21 In the latter case, the regions of negative potential energy are found by using a test charge to scan the potential energy surface produced by solvent molecules. The present methods of theoretJlca1description of excess e k e trons in polar media q u i r e knowledge of the electron-molecule interaction potential. The potential should be fairly reliable and simple enough for very frequent evaluation in a simulation program. Such an explicit form of the interaction potential, V,has been derived only for the electron-water and for the e1ectro"monia system24(in the latter case, however, the potential is limited to the truncated Coulombic term). The present paper is devoted to the construction of the interaction patential between an excess electron and the methanol molecule OR the basis of the electron-molecule scattering theory and the pseudopotential theory. Our approach is based on a general method of constraction of the pseudopotentials for excess electrons in polar media which has been delineated in ref 22. In the following &s we describe tbe individual terms of the pssvdapotential and the methods which we have employed in the calculations. (15) Wallqvist, A.; Tkirumalai, D.; Beme, B. J. J. Chem. Phys. 1987.86, 6404. (16) Wallqvist. A.; Martyna, G.; Berne, B. J. J. Phys. Chem. 1988, 92, 1721. (17) Bamett, R. N.; Landman, U.; Cleveland, C. L.;Jortner, J. J. Chem. Phys. 19811, 88, 4429; J . Phys. Reo. Lctr. 1987, 59, 811. Landman, U.; Bameu, R.N.;Cleveland, C. L.;Scharf, D.; Jortner, J. J. Phys. Chem. 1987, 91, 4890. (18) Barnett, R. N.; Landman, U.; Nitzan. A. Phys. Rev.Lerr. 1989.62, 106, J. Chem. Phys. 1989,90,4413. Motakabbir, K. A.; Schnitker, J.; Rossky, P. J. J. Chem. Phys. 1989, 90, 6916. Webster, F.; Rossky, P. J.; Friesner, R. A. Comput. Phys. Commun., in press. (19) (a) Chandler, D.; Wolynes, P.G. J. Chem. Phys. 1981, 74,4078; 75, 1343. Chandler, D. J. Phys. Chem. 1984, 88, 3400. (b) Sprik, M.; Kelin, M. L. J . Chem. Phys. 1987,87, 5987; 1988,89, 1592. (c) Barnett, R. N.; Landman, U.; Nitzan, A. J. Chem. Phys. 1988,89, 2242. (20) Hilczer, M.; Bartczak, W. M.; Sopek, M. J . Chem. Phys. 1986,85, 6813. Hilczer, M.; Bartczak, W. M.; Kroh, J. J . Chem. Phys. 1988,89,2286. (21) Schnitker, J.; Rossky, P. J.; Kenney-Wallace, G. A. J. Chem. Phys. 1986,86, 2986. Motakabbir, K. A.; Rossky, P. J. Chem. Phys. 1989, 129, 253. Hilczer, M.; Bartczak, W. M.; Sopek, M. Radiar. Phys. Chem. 1989, 36, 199. (22) Schnitker, J.; Rossky, P. J. J. Chem. Phys. 1987, 86, 3462. (23) Barnett, R. N.; Landman, U.;Cleveland, C. L. J. Chem. Phys. 1988, 88, 4421. (24) Sprik, M.; Impley, R. W.;Klein, M. L . J. Chem. Phys. 1985, 83, 5802. Marchi, M.; Sprik, M.; Klein, M. L. J. Phys. Chem. 1988, 92, 3625.
0 1992 American Chemical Society
The Journal of Physical Chemistry, Vol. 96, No. 6, 1992 2737
Electron-Methanol Pseudopotential 11. Contributions to the Electron-Methanol Pseudopotential The effective electron-molecule potential is usually expressed as V(?,(8j))= V C VR VEX Vp (1) where P is the electron coordinate and I&) stands for the set of coordinates of molecular nuclei. The first term, Vc, corresponds to the Coulomb potential. The repulsion term, VR, is connected with the Pauli principle, which requires orthogonality of the wave function of the excess electron to all the occupied orbitals of the molecule. The exchange potential, VEX,takes into account the indistinguishability of the excess electron and the molecule electrons. The polarization potential, Vp, results from the molecule polarization induced by the electron charge. 1I.A. Electrostatic Potential. The potential Vc includes the static nuclear, VCN,and the static electronic, VCE,terms. The static nuclear potential has the form
+
+
I(*)
+
where Zi is the charge of the ith nucleus. The static electronic potential is defined as (3)
In the SCF approximation, the electronic density, p, is related to the set of the molecular orbitals, J/,by P(7d = 2cJ/a*mJ/a(~,)
Figure 1. The coordinate system for the methanol molecule and the spatial directions for Figures 4, 6, and 12. The axes I,, 12, and I3 are defined by the OH, CO, and CH bonds, respectively. Axis l4 is the bisecting line of the HCH angle. (+) and (-) correspond respectively to the positive and negative directions from the central atom on a given axis.
Gaussian function^.^' This method of calculation requires, however, so much computer time that it cannot be applied to the computer simulation of an excess electron in molecular matrices. The charge density of a methanol molecule is, fortunately, concentrated in the close vicinity of the nuclei (cf. Figure 7), and it is possible to approximate the total electrostatic potential by
(4)
VC(P,(&))= -E-
(1
The last formula can be expressed in terms of the normalized atomic orbitals, 4, as ~ ( 7 ,= ) cePij4i*(PiMj(ii) i
J
(5)
where pij are the elements of the one-electron density matrix, ji For the Gaussian basis the orbital 4i is a linear combination of the primitive Cartesian Gaussian functions
4i(Pi) =
CCwiKJF1)
(6)
J !
with K J P ~ ) = exp(-e,ir12)xl’ylJzlK (7) where 7, = (xl,yl,zi),e,; is an orbital exponent, and the integers I, J , and K determine the type of the orbital di as I + J + K = 0, 1, or 2 for s, p, and d orbitals, respectively. Thus, the electronic potential, VcE, can be evaluated as
In tiis exprpion the primitive Gaussians, K~ and K ~ are ~ , centered at Ri and Rj, respectively, and Pli = 7, - R,. In the present paper the density matrix, 6,is calculated using the computer program MICROMOL v with the UHF procedure and the contracted (9s45pz/4sz)Gaussian basis set of double-{ quality worked out by Dunning.z5 The geometry of a methanol molecule is described by the gas-phase values of the intramolecular bond lengths rOH= 0.9451 A, roc = 1.4246 A, and rCH= 1.0936 A and by the angles &OH = 108.53’, PoCH = 110.36’, and BHCH = 108.63’. The same molecular geometry was assumed in the computer simulations of liquid m e t h a n 0 1 . ~ ~ - ~ ’ ~ ~ ~ In order to compute VcE from eq 8, it is necessary to evaluate the integrals involving all the combinations of the primitive (25) Dunning, T. H. J . Chem. Phys. 1970,53, 2823. (26) Jorgensen, W. L. J . Am. Chem. SOC.1981, 103, 341. (27) Haughney, M.; Ferrario, M.; McDonald, I. R. J . Phys. Chem. 1987, 91, 4934. (28) Williams, D. E. J . Comput. Chem. 1988, 9, 745. (29) Pilinkis, G.; Hawlicka, E.; Heinzinger, K. J . Phys. Chem. 1987, 91, 4334. Hawlicka, E.; PilinkBs, G.; Heinzinger, K. Chem. Phys. Leu. 1989, 154, 255. (30) Jorgensen. W. L. J . Phys. Chem. 1986, 90, 1276.
i
4i
li - Ri(
(9)
where the summation runs over “sites” on the molecule (usually, but not necessarily, nuclei) and qi is the effective point charge at the ith site. We have considered a number of different point charge models for the methanol molecule.26M The electrostatic potential obtained from the ab initio calculations is well fitted by the set of the atomic monopoles proposed by Williamsz8 and by the H1 model of Haughney et al.z7 The former model is, however, fully adequate only for the staggered conformation because it assumes a positive charge on the methyl hydrogen in the trans position and negative charges on the two remaining hydrogen sites of the CH3group. This model is not satisfactory for other molecular conformations which can appear in the computer simulation of liquid methanol. Therefore, in the present calculations we have chosen the H1 model for which the partial charges assigned to the particular sites are as follows: oxygen, qo = -0.728 lei, hydroxyl hydrogen, qH = 0.431 lei, carbon, qc = 0.297 lei, and each methyl hydrogen, qHc = 0. Comparisons of the electrostatic potentials for the methanol molecule obtained on the basis of the ab initio calculations (eqs 2 and 8) and from the H1 model (9) are presented in Figures 2, 3, and 4. Figure 1 defines the coordinate system of the plots and the spatial directions (from I , to 14) which are considered in certain figures of this paper. The axes I ] , 12, and 1, are determined by the directions of the OH, CO, and C H bonds, respectively. The axis l4 is defined as the bisecting line of the HCH angle. The isoenergy contour maps in the planes y = 0 and z = 0 are plotted in Figures 2 and 3, respectively. The maps of the ab initio (parts a of the figures) and the H1-model (parts b) potentials Vc are quite similar except for the space region in the vicinity of the nuclei. In this region, however, the repulsion potential, VR, dominates over Vc and the sums Vc + VR calculated for the a b initio and H 1 electrostatic potentials are almost identical. This is documented in Figure 4, where the two potentials Vc and their sums with VR (see eq 14) are shown along the axes I , to 14. The plots of Vc V, along l2 to I, prove that the strong repulsion around the carbon atom covers even the differences between the potentials Vc which are caused by the lack of the point charges
+
(31) Webster, B. C.; Hilczer, M.; Ramos, M. J.; Carmichael, I. J . Chem. SOC.,Faraday Trans. 2 1985, 81, 1761. Ramos, M. J.; Webster, B. C. J . Chem. SOC.,Faraday Trans. 2 1983, 79, 1389.
2738 The Journal of Physical Chemistry, Vol. 96, No. 6,1992
Hilczer et al. d i s t o n c ~from
c
-41,
a atom
,
I
diatance from C ntom, b,
I
,
,
,
~
JL
\io/,
/, , ,
/
-6 .6
,
,
I
.3
0 3 distanm Ira C otom ,
A
,I[,
6
1
, , .6
, ,
,
.3
J
!:
,/: 0
distonce from
, , , , , , 3
1
6
C atom, A
Figure 4. Comparison of the electrostatic potentials Vc calculated from the ab initio wave functions and from the HI model and the sums of V, with the repulsion potential, VR (414), in four spatial directions I, to 1, (see Figure 1): Vc (ab initio), (-*-); V ,(HI model), (---); Vc (ab initio) + VR, (- -); V, (HI model) + VR, (-).
Figure 2. Electrostatic potential of interaction hetwecn an electron and the methanol molecule in the plane y = 0. The upper part (a) of the figure shows the contours calculated from the ab initio wave functions, the lower part (b), the contours from the electrostatic part of the H1 model of methanol (ref 27). The contour lines labeled -6 to 6 correspond tothevalues-0.5,-0.3,-0.2,-0.1,-0.05,~.02,0,0.02,0.05,0.1,0.2, 0.3, and 0.5 eV, respectively.
H and C nuclei, respectively. The switching function removes very narrow attractive singplarities of the potential V at the hydroxyl hydrogen and carbon positions. The same switching function has been applied to provide a smooth shape of the sum Vc V, shown in Figure 4. 1LB. Rcpllbbn POdathL The repulsion potential of a lnolscuie can be approximated by a sum of repulsion terms centered at the nuclei*
+
where Ai, Bi, and ai are adjustable parameters. Schnitker and Rossky,22using a method consistent with the formalism of the scattering theory and pseudopotential theory, have proposed another expression for VR VR(FJ&I)
-Eea+a(3l+a(Fl)
dF1
(12)
where ea is the energy eigenvalue associated with the molecular orbital $a. Expanding +a into the series of the functions #j, they have obtained the following formula for VR
VR(P,{&)
= ccjdj(7j) J
(13)
where ij = i - f i j ; the sum runs over the spherically symmetric atomic orbitals 4,. We have applied the Schnitker and Rmky method for the case of tbe methanol molecule. The wave functions, +u, and the eigenvalues, eu, which are necessary to evaluate the coefficients C, in eq 13, have been obtained from the ab initio calculations with the contracted Gaqqian basis set (cf. section 1I.A). Using the analyticaLform of the primitive Gaussian functions (7), the expression for VR can be written as Figure 3. Electrostatic potential of interaction between an electron and the methanol molecule in the plane z = 0. (See legend to Figure 2.)
on the methyl hydrogens in the H1 model. The Coulomb contributionsto the total pseudopotential, V (eq l), associated with the positive point charges axe, similarly as in ref 22, multiplied by switching functions of the form S(lPil) = 1PiI2(3ru- 21Pi1)/rU3, for lFil Iru
(10)
= 1, for lPil > ru where P, = P - d,,d, = dH or dc, and r,, = 0.3 and 0.05 A for
where the vectors dj describe the positions of molecular nuclei and the parameters B, and e,,, are collected in Table I. Contours of the repulsion potential in the plane defined by the positions of the H,0,C,and Hc atoms 01 = 0 plane) are shown in Figure 5 . The potential vs distance along the OH (II axis) and CO (12 axis) bonds is plotted in parts a and c of Figure 6, respectively. Additionally, the figures compare the potential (14) with the potential VRBcalculated for methanol from the relation proposed
The Journal of Physical Chemistry, Vol. 96, No. 6,1992 2739
Electron-Methanol Pseudopotential TABLE I: Parameters for the Repulsion Potential, VR (in Atomic Units)
-4
distance f r o m 0 atom , -2 0 2
A 4
~
i
oxygen
1
nI 6
hydroxyl hydrogen
2 3 4 5
1 1 1 3
carbon
6 7
1 6
8 9 10 11
1 1 1 3
site
methyl hydrogen
12
1
Brl 8.6744 15.9243 25.4684 34.4013 31.4224 3.1122 19.0711 3.5650 1.1374 0.3428 0.5839 0.6751 0.0123 4.7721 8.8064 14.2031 19.1812 19.5473 1.8051 9.0997 2.0914 0.7039 0.3601 0.6134 0.7092 0.0748
%I
78 16S400 1175.8200 273.1880 81.1696 27.1836 3.4136 9.5322 0.9398 0.2846 19.2406 2.8992 0.6534 0.1776 4232.6100 634.8820 146.0970 42.4974 14.1892 1.9666 5.1477 0.4962 0.1533 19.2406 2.8992 0.6534 0.1776
ii -2 0 2 distance from C atom ,
-4
4
1
Figure 6. The repulsion potential, V, (parts a and c), and the exchange potential, VEX(parts b and d), vs distance along the I, and l2 axes, respectively. V, is calculated as follows: from eq 14 (-); from eq 15 with the ab initio electron density, p (- -); and from eq 15 with p fitted by eq 17 (- -). VEXis calculated from eq 19 with the ab initio p (- -) and with the approximate p (17) (-).
-
TABLE II: The Optimized Fitting Parameters for the Approximated Form (17) of the Electron Density of the Methanol Molecule (in Atomic Units) oxygen
-5
0
5
hydroxyl hydrogen
z, A Figure 5. Contours of the e--methanol repulsion potential (eq 14) in the plane y = 0. The labels 1-9 correspond to the values lV5,lo4, 0.05,0.2,1, 3, and 10 eV, respectively.
carbon
They modeled the core repulsion by a “local by Barnett et kinetic energy” term
methyl hydrogen
VRB(?,{Aj})= 0.5[3~’p(7)]~/’ (15) This term represents the increase in the kinetic energy of the exelectron due to the orthogonality constraint between the excess electron wave function and the molecular orbitals. The potential VRBobtained using the density, p, evaluated from the a b initio wave functions, is shown in Figure 6a,c as the solid lines with circles. Functions 14 and 15, in spite of rather large differences in the regions “inside” of the molecule, agree reasonably well in the regions of importance, e.g., the regions accessible by an excess electron. Moreover, the absolute minimum of the total pseudopotential, V (see eq 1 and Figure 12a), does not change considerably if we use (15) instead of (14)to describe the repulsion contribution to V. While using eq 15 in practical computations, it is comfortable to have a simple approximate expression for the electron density, p. Such a fitting for the methanol molecule is described in the following section. The dashed curves in parts a and c of Figure 6 present the potential VRB vs the distance along the II and f 2 axes, respectively, obtained for the density, p, which has been calculated from the approximate expression (17). II.C. Approximation for the Electron Density. The molecular electron density, p, can be approximated by a combination of a modest number of Gaussian functions centered at the nuclei of
0.4830 1.6819 36.5526 156.8656 0.4701 3.5021 0.2221 1.1171 27.3938 139.0907 1.4115
3.2125 3.7170 1.3646 0.3659 0.2337 0.2940 2.0728 2.6455 1.4478 0.1379 0.8422
a molecule. The fitting is performed by optimization of the following functional32
where
p*
is an approximation to p in the form
p*(q
DjGi(3
(17)
i
9 stands for the normalized Gaussian function centered at a point 4 ~ ~ ( =7 (ai/7r)3/2 ) exp[-ai(F - di)2]
(18) Equation 16 expresses the least-squares fitting of the electron density to p* with the function l/lF1- 721as a weighting function. The functional W is minimized with respect to the linear parameters, b , and nonlinear parameters, ai. In the present calculations the electron density, p , of a methanol molecule is approximated by a combination of the Gaussian functions centered at the oxygen (four), carbon (four), hydroxyl hydrogen (two), and each methyl hydrogen (one). The optimized (32) Hall, G.G.;Smith, C.M. I n f . J . Quantum Chem. 1984, 25, 881.
2740 The Journal of Physical Chemistry, Vol. 96, No. 6, 1992
Hilczer et al. - 5 -
0
-5
= , A
5
Figure 8. Contours of the exchange potential (eq 19) in the plane y = 0. The electron density, p, was fitted by eq 17. The labels 1-8 correspond to the values -lo", -lo4, -lo-*, -0.05, -0.2, -1, and -3 eV, respectively.
0
-5
5
z, A Figure 7. Contours of the logarithm (base 10) of the electron density, p (the atomic units), in the plane y 3: 0. The upper part (a) is calculated from the ab initio wave functions, the lower part (b), from the Gaussian approximation (17). The labels 1-9 correspond to the logarithmic values -2, -3, -4, -5, -6, -7, -8, -9, and -10, respectively.
values of the parameters a, and b, are given in Table 11. Accuracy of the fitting is characterized by the value of Wat the optimum (0.0043 hartree) as compared to the self-energy Wo= 96.1382 hartree, which represents the value of functional 16 for p* E 0. In Figure 7a we show the electron density contours calculated from the ab initio wave functions. Figure 7b presents the electron density obtained from eq 17. Both maps are plotted in the plane y = 0. ED. Exchnge Potentid. In the electron-molecule scattering theory, VEXis usually approximated by local, energy-dependent, free-electron-gas exchange p ~ t e n t i a lof~ ~the, ~form ~ (in the atomic units) VE~.(~?(fii~) = -(2/*)kFF(1) (19) where
+ kF2)'l2/kF
(21)
kF(3 = [ 3 ~ ~ p ( 7 ) ] ' / ~
(22)
1 ( 3 = (kz + 21
1 is the first ionization potential of a molecule and k is the
As can be seen from Figure 12a, the exchange contribution to the total potential, VI is quite important in the vicinity of the absolute minimum of V. LIE Pohuization PoteatkL The preaence of an excess electron in the vicinity of a molecule causes a distortion of the charge cloud of the molecule. This effect is not accounted for in the discussed terms of the electron-molecule potential which are calculated from the wave functions of the ground state of an isolated molecule. Thus, an additional polarization term has to be included in the total potential, V. This term is usually limited to the interaction between the electron and the induced dipole. Low energies involved in the electron solvation justify the adiabatic approximation to Vp The simplest asymptotic form of the polarization potential, V,, is given by Vps(r) = -a0/2#
-
(23)
where a,, is the spherical dipde polarizability of the molecule. To avoid the divergence of V, at r 0, eq 23 is modified to the form Vps(f)= -a&)/2r4
(24)
f(r) = I - exp[-(r/r$]
(25)
where
is the switching function with a cutoff radius, r, chosen on the condition that the polarization potential has negligible strength within the molecular charge cloud. Hence, r, is often taken as the size of a molecule. Such a form of the potential has been widely used in the electron scattering the0ry.~3'3 This expression has been also employed in path-integralsimulations of the hydrated electr~n.~~J~~~~ The adiabatic polarization potential for selected molecules has been evaluated by ab initio quantum chemical methods.39 Before employing it in quantum simulations, the ab initio potential Vp has to be fitted to a simple mathematical Concerning the quality of such a fitting, the large numerical effort required to evaluate the ab initio potential seems to be rather superfluous. In the present calculations we consider a nonspherical shape of the methanol molecule. Thus,the polarization potential energy is expressed as
wavenumber of the scattering electron. This approximation to V , is quite good at very low impact e n e r g i e ~ . 3 Similarly ~ ~ ~ ~ as in ref 22, we take k = 0. The ionization energy is assumed as 7.6971 eV and corresponds to the value of 1 for a methanol molecule in the dielectric cavity.36 Figure 8 presents the contours of the exchange potential in the plane y = 0 calculated for the electron density, p, as fitted by eq 17. Parts b and d of Figure 6 show VEXalong the OH and CO bond directions, respectively. The figure shows the comparison of the exchange potential calculated for the density, p, obtained from the ab initio wave functions and the potential obtained using the approximation (17).
where 2 = ( f - d13is the electric4eld due to the excess electron, Q is the polarizabilitytensor, and R describes the position of the center of mass of the methanol molecule. Three principal components of the polarizability tensor are taken asMa,= 4.09
(33) Hara, S. J . Phys. SOC.Jpn. 1967, 22, 710. (34) Collins, L. A.; Norcross, D. W. Phys. Reu. A 1978, 18, 467. Morrison, M. A.; Collins, L. A. Phys. Reu. A 1978, 17, 918. (35) Riley, M. E.; Truhlai, D. G. J . Chem. Phys. 1975, 63, 2182. (36) Medina-Llanos, C.; Agren, H.; Mikkelsen, K. V.; Jensen, H. J. J . Chem. Phys. 1989, 90,6422.
(37) Burke, P. G.; Chandra, N. J . Phys. E 1972, 5, 1696. (38) Jain, A. J . Chem. Phys. 1983, 78, 6579. (39) Dougkss, C. H.; Weil, D. A.; Charlier, P. A.; Eades, R. A.; Truhlar, D. G.; Dixon, D. A. In Chemical Applications of Atomic and Molecular Electrostatic Potenfiafs;Politzer, P., Truhlar, D. G., Eds.; Plenum: New York, 1981; p 173.
V,(j,fi) = -Y2EaE
(26)
The Journal of Physical Chemistry, Vol. 96, No. 6, 1992 2741
Electron-Methanol Pseudopotential
z. A
0
-5
z ,
5
a
Figure 9. Contours, in the plane y = 0, of the polarization contribution (eq 26 multiplied by eq 25) to the electron-methanol interaction potential. The labels 1-7 correspond to the values -0.02,-0.05,-0.1, -0.2, -0.4,-0.8, and -1 eV, respectively.
Figure 11. Total potential, V, in the planes x = 0 (upper parts, a) and z = 0 (lower part, b). The labels -8-8 correspond to the values -0.6,
-0.5, -0.4,-0.3, -0.2,-0.1,-0.05,-0.02,0,0.02,0.05,0.1,0.2,0.3,0.5, 1, and 2 eV, respectively.
A
distance f r o m 0 atom,
2 . -
101 -10
\ '
'
'
'
0 distance ,
'
'
A
'
i
10
Figure 10. Total potential, V, of the electron-methanol interaction (eq 1) in the plane y = 0 (upper part, a ) and in the plane perpendicular to the I, axis and passing through the absolute minimum of V(lower part, b). The labels from -10 to 8 correspond to the following values: -1.4, -1, -0.7,-0.5, -0.4,-0.3,-0.2,-0.1, -0.05,-0.02,0,0.02.0.05,0.1,0.2, 0.3,0.5,1, and 2 eV, respectively.
A3, a2= 2.65 A3, and a3 = 3.23 A3, The principal axes of the molecule are defined by the angles: L(&,,C-O) = 21.07'; L(&,,&H) = 50.4O; ?i2 ICOH plane. The potential is multiplied by the switching function of the form (25) with the cutoff radius r, = 2.059 A. This value of r, corresponds to the distance between the center of mass of the molecule and the most distant methyl hydrogen plus the hydrogen Bohr radius. Contours of the potential Vp in the y = 0 plane are plotted in Figure 9. The minima of the polarization energy, shown in the figure, are placed on the first principal axis (Z,) of the methanol molecule. The value of Vp at these minima is equal to -1 .OS1 eV. The assumed form (26) of the polarization potential makes the total potential, V (eq l), more directional as compared to the spherically-symmetric potential (24) and facilitates the approach of the excess electron to the methyl group and the hydroxyl hydrogen. The value of V, at the absolute minimum of the total (40)Applequist, J.; Carl, J. R.; Fung, K.-K. J . Am. Chem. Sor. 1972, 94, 2952.
-0
-4 0 distance from C a t o m ,
4
A
8
Figure 12. The plots of the total electron-methanol potential, V, and its components vs distance along the OH (upper part, a) and CO (lower part, b) bonds: V, (- -1; Vc (eq 9),(-); VR (eq 14),(- - -); VEX(eq 1% V, (eqs 26 and 25), (e-);
(-e-).
potential, V (which w u r s in front of the hydroxyl hydrogen, cf. Figure 12a), is lower than the corresponding value of the spherically-symmetricpotential, V,, by about 12%. At the point behind the carbon atom [the minimum shown in Figure 12b), Vp is lower than Vp, by about 17%. 111. The Total Potential
The total potential, V,expressed as the sum of the static (eq 9 with the qH and qc contributions multiplied by the switching
2742 The Journal of Physical Chemistry, Vol. 96, No. 6, 1992 terms (lo)), repulsion (eq 14), exchange (eq 19), and polarization (eq 26 modified by (25)) terms is displayed in Figures 10-12. Figure 1Oa presnts the contour map of Vin the plane y = 0. The minimum of Valong the II axis (cf. Figure 1) at a distance r,,, = 2.24 A from the oxygen atom is -1.61 eV. The attractive potential surrounds the hydroxyl hydrogen as well as the whole CH, group. In Figure lob we show the isoenergy contour map in the plane passing through r = rminand perpendicular to the OH bond. Taking into account the symmetry of the attractive potential around the H nuclei, we can expect that the minimum at r = rmi,corresponds to the absolute minimum of V. Parts a and b of Figure 11 show the total potential, V, in the planes x = 0 and z = 0, respectively. In parts a and b of Figure 12 the potential Vand its parts are plotted vs distance along the OH and CO bond directions. The contributions to the absolute minimum of Y (Figure 12a) from the particular terms of the potential are Vc = -1.53 eV, V, = 0.81 eV, V, = -0.24 eV, and Vp = -0.65 eV. For comparison, the contributions to the minimum at the backside of the carbon atom (Figure 12b) are -0.31 eV, 0.09 eV, -0.003 eV, a d -0.12 eV, respectively. It is interesting to compare the electrowmethanol potential with some electron-water pseudopotentials proposed in the literature. The potential derived by Schnitker and R o ~ s k yis~composed ~ of the Vc, V,, and Vp terms. The exchange potential, although discussed by the authors, has been omitted from their calculations. On the other hand, Barnett et al?, emphasize the importance of YE,,particularly in the vicinity of the hydrogen sites. In our calculations the exchange effect is negligible around the CH, group but cannot be neglected in the vicinity of the absolute minimum of the pseudopotential V. Wallqvist et al.lSJ6emphasize the role of the polarization term and introduce to the path-integral simulationsof eaq-the manybody polarization effects. The present calculations of the model electron-molecule pseudopotential show that the simple spherical approximation (24) to Vp (employed in refs 15, 16, and 22) is insufficient in the case of the methanol molecule. The depth of the minimum of the present potential is similar to the depth obtained for the e--H,O interaction in ref 22, but it is larger as compared to that of refs 15 and 16. On the other
Hilczer et al. hand, the optical absorption spectrum of the hydrated e l e c t r ~ n ~ ~ ~ , ~ simulated with the former potential22is by about 0.7 eV blueshifted in comparison with both the experimental data and the results of ref 16. Moreover, the positions of the absorption bands measured for e;in methanol and water are quite close (the former is shifted only by about 0.15 eV toward higher energies). The relation between the depth of the electron-molecule pseudopotentials and the position of the G-absorption bands in various solvents is, however, not very simple. The position of the maximum of the band is strongly affected by the average local structure of solvent molecules in the vicinity of the excess electron. The electron-water radial pair-correlation functions found from the path-integral simulations of eq- show six bond-oriented molecules in the fmt hydration layer of the electron.'lb-l6 The same solvation structure of e,- is also postulated by Kevan4' on the basis of the ESR measurements. Similar experiments4' for the trapped electrons in a methanol matrix give 4-fold coordination and dipoleoriented solvent. A lower value of the average coordination number and leas favorable orientationsof methanol molecules as compared to water force us to expect that the methanol traps are more shallow than the aqueous traps. As a consequence,we expect that the e; absorption band calculated with the present potential will be shifted toward low energy in comparison with the spectrum of e,; obtained with the potential of ref 22. The prcsent potential,limited to the electrostatic and repulsion contributions,can be d y applied in the analysis of the preexisting electron traps in a computer-simulated methanol matrix. The absolute minimum of the abridged potential is about -0.86 eV at a distance r = 2.48 A from the oxygen atom along the OH bond direction. The addition of the successive polarization contribution gives the minimal value of the resulting three-term potential as -1.4 eV at r = 2.32 A.
Acknowledgment. This work has been supported by the Institute of Applied Radiation Chemistry, LM2, Poland as a part of Project CPBP 01.19. Registry No. Methanol, 67-56-1. (41) Kevan,
L.Radiat. Phys. Chem. 1981, 17, 413.
