ELSEVIER

29 August 1997

Chemical Physics Letters 275 (1997) 377-385

CHEMICAL PHYSICS LETTERS

All-quantum description of molecules: electrons and nuclei of C 6 H 6

Rafael Ramfrez a, Joachim Schulte b,1, Michael C. Bi3hm b a lnstituto de Ciencia de Materiales, Consejo Superior de lnvestigaciones Cient~cas, Cantoblanco, E-28049 Madrid, Spain

b lnstitutfiir Physikalische Chemie, Physikalische Chemie lIl, Technische Itochschule Darmstadt, D-64287 Darmstadt, Germany

Received 4 June 1997; in final form 18 June 1997

Abstract

Feynman path integral (FPI) quantum Monte Carlo simulations have been combined with electronic ab initio calculations of Hartree-Fock (HF) type to achieve an all-quantum description of the benzene molecule. The linking of these two quantum approaches allows the consideration of the quantum character of the atomic nuclei and electrons. The FPI formalism has been employed to generate 6000 different nuclear configurations of the benzene molecule which are populated in thermal equilibrium (canonical ensemble statistics). In a second step the ab initio HF Hamiltonian has been used to calculate (electronic) expectation values as ensemble averages over these configurations. The influence of the nuclear dynamics on the kinetic and potential energy of the electronic Hamiltonian is discussed. The spatial nuclear degrees of freedom of benzene are largely determined by quantum fluctuations which exceed the thermal fluctuations even at room temperature. The importance of the quantum delocalization of the nuclei is emphasized on the basis of calculated radial and angular distribution functions. © 1997 Elsevier Science B.V.

1. Introduct ion

The often highly sophisticated techniques of molecular quantum chemistry have grown to become an important tool for comprehensive investigations of electronic structure problems. The merits of the different approaches have been summarized in a large number of review articles and text books [1,2]. With the exception of some attempts published re- cently the quantum degrees of freedom of the atomic nuclei have been neglected in the evaluation of electronic expectation values [3-7]. In short, quan- tum chemical electronic structure methods consider

i Present address: Bruker Analytische MeBtechnik GmbH, Sil- berstreifen, D-76287 Rheinstetten, Germany.

the nuclei as space-fixed classical particles. The limitations of such an approximation are clearly seen m the temperature dependence of chemical shifts or nuclear quadrupole resonance (NQR) frequencies. These electronic expectation values are a function of temperature; they reflect the nuclear dynamics. A "hybr id" method of considering the dynamics of the nuclear framework is the Car-Palr inel lo (CP) method which combines a quantum description of the elec- trons with classical dynamics of the nuclei [8]. Both degrees of freedom are taken into account simultane- ously. In molecules with light atoms the non-consid- eration of the quantum character of the nuclei cannot be justified a priori, In a series of publications we have demonstrated the importance of the quantum character of the nuclei even at temperatures T a p -

0009-2614/97/$17.00 © 1997 Elsevier Science B.V. All rights reserved. PII S0009-26 14(97)00754-9

378 R. Ramfrez et al. / Chemical Physics Letters 275 (1997) 377-385

proaching room temperature (RT) [9-14]. In these studies we have employed the powerful Feynman path integral quantum Monte Carlo (FPI QMC) tech- nique of quantum statistical physics [15-18]. Our simulations have shown the large spatial fluctuations of light atoms as well as the importance of tunneling effects. Due to its flexibility the FPI method allows the transition from a fall quantum description of the nuclei to a classical one as adopted in the majority of molecular dynamics (MD) simulations. Furthermore, the method can be used to extend the simple har- monic oscillator (HO) approximation of the vibra- tional problem to more elaborate schemes beyond the HO boundary. For discussions of this topic see Refs. [12,13].

In a recent publication [5] we have used the results of a FP1 QMC simulation which considers the quantum degrees of freedom of the nuclei as input for a succeeding electronic structure approach. In this second step we have calculated electronic expec- tation values as ensemble averages over the nuclear configurations generated by the quantum Monte Carlo formalism. In the framework of the Born-Op- penheimer approximation (BOA) the nuclei are free to thermally populate (with canonical ensemble sta- tistical weights) any state of the vibrational problem. In Ref. [5] we adopted the simple Pariser-Parr-Po- pie (PPP) 7r Hamiltonian [19,20] as well as the two-parameter Hubbard (Hu) approach [21] as com- putational tools in the electronic structure part of our all-quantum approach. In the present work we com- bine a FPI QMC formalism for the nuclear degrees of freedom with an ab initio Hartree-Fock (HF) Hamiltonian to evaluate electronic expectation val- ues. We analyze the influence of the nuclear dynam- ics on ab initio based electronic quantities. We have adopted the benzene molecule C6H 6 as a model system. The electronic expectation values of this xr system have been calculated as "ensemble averages" sampled over 6000 different nuclear configurations which have been generated via a FPI simulation. In the statistical approach we have adopted a model potential V(R) for the interatomic interaction. This model potential has been designed to reproduce the experimental geometry of C6H 6. V(R) should ap- proximate the potential energy surface of the ab initio approach which is constructed on the basis of the total electronic energy Eto t. Eto t is defined by the

kinetic energy of the electrons Ekin, the electron-core attraction Eel,nuc, the electron-electron repulsion Eel and the nuclear repulsion Enu c. To discriminate the single-configuration quantities E i of the electronic Hamiltonian from the ensemble averages we have used the labels ffSi (i = kin, nuc, el, el,nuc, pot, tot) to denote the ensemble averages. This letter should be accepted as our first step towards temperature-de- pendent (molecular) ab initio quantum chemistry, i.e. quantum chemistry based on an all-quantum treat- ment of the molecule under consideration. In our approach the quantum character of the electrons and the nuclei are both taken into account. Other theoret- ical contributions with this intention have been men- tioned above [3,4,6,7]. In contrast to these works we present a microscopical fragmentation of changes in the covalent bonding as a response to the spatial fluctuations of the atomic nuclei. Such a microscopi- cal access is missing in other all-quantum studies of molecules.

The organization of this Letter is as follows. The theoretical background of the methods used and the computational conditions are commented in Section 2. In Section 3 we discuss the results of the all- quantum study of benzene. We evaluate ensemble averaged expectation values of the ab initio Hamilto- nian at temperatures T of 50 and 750 K. The letter ends with a short resumC

2. Theoretical techniques and computational con- ditions

The general background of the Feynman PI for- malism and its range of applicability for fermionic and bosonic problems is well documented in the literature [ 15-18,22]. In recent publications we have adopted this technique both for bosonic [9-13] and fermionic [23,24] systems. The FPI method makes use of an isomorphism between a single quantum particle (here each nucleus p from 1 to P of C r H 6) and a cyclic chain of N classical particles (= beads). This isomorphism allows the partition function Z of the quantum ensemble to be approximated by a classical one which can be studied by the methods of classical simulation techniques such as MD or MC. By taking a sufficiently large number of beads an accurate evaluation of the quantum partition function

R. Ram[rez et aL / Chemical Physics Letters 275 (1997) 377-385 379

is possible. In the present FPI implementation the quantum particles interact through an effective many-body potential V(R) which has been taken from Ref. [25]. To approach the experimental bond lengths of benzene [26] we have changed the rcc value of 152.2 pm of Ref. [25] (rcc symbolizes the length of the C - C bonds) into rcc = 139.7 pm. All other intramolecular elements in the definition of V(R) correspond to the design of Ref. [25]. With this parametrization of V(R) we derive C - C and C-H bond lengths of C6H 6 of 140.7 and 109.2 pm. Both numbers are close to the experimental values of 139.7 and 108.4 pm [26]. The optimized ab initio bond lengths in the 3-21 G basis set adopted amount to 138.5 and 107.2 pm. The agreement between both computational results (FPI approach versus ab initio HF calculation) is sufficient. Therefore, it can be assumed that V(R) is a trustworthy zeroth order approximation to the potential energy surface of the ab initio Hamiltonian.

The partition function Z of the quantum ensemble of P atomic nuclei at temperature T has been evalu- ated through a discretization of the so-called thermal density matrix p along a cyclic path composed of N steps ( = number of beads). N represents the number of time-slices in this discretization procedure. Their width amounts to flh/N with /3 = (kBT) -1. h de- notes the Planck constant and k B the Boltzmann constant. To derive Z with sufficient accuracy the so-called short-time approximation has been adopted. The effective potential experienced by the beads in the cyclic chain amounts to 1/N of V(R). This "renormalization" of the true potential V(R) in the determination of thermal expectation values supports the penetration of the nuclei ( = atomic cores with their electron clouds) into regions of higher potential. The electronic response of this quantum mechanical penetration is derived via a succeeding ab initio ensemble averaging of expectation values of the corresponding electronic Hamiltonian. A classical simulation of the nuclei at finite temperatures T is obtained when setting N = 1. The atomic nuclei within each of the P classical chains at time-slice j are "harmonically" coupled to their j + 1 and j - 1 images. The coupling between chains via the poten- tial V(R) requires the same imaginary time-slice argument j, a computational condition which takes into account instantaneous interactions.

To derive the T ~ 0 K properties of C6H 6 via FPI simulations we have adopted the classical Metropolis MC sampling technique [27]. The results (= nuclear configurations) of FPI simulations of quantum type at temperatures of 50 and 750 K have been used as input for succeeding ab initio HF calculations. The T = 50 K properties of benzene are mainly deter- mined by zero-point vibrations. Non-ground-state contributions are of minor importance only. To cal- culate the thermal properties of C6H 6 5 X 104 MC steps have been performed. 2 × 104 quantum paths have been taken into account for the system equili- bration. The number of time-slices N has been chosen to obey the relation NT = 3000 K, a statisti- cal convergence criterion which guarantees FPI re- sults of high quality [18]. Under low-temperature conditions this choice generates a set of 3 × l0 6

different nuclear configurations of C6H 6. In the high-temperature simulation a set of 2 X l05 config- urations occurs. 6000 C 6 H6 configurations from each of these sets have been adopted for the calculation of ensemble averaged electronic expectation values via ab initio HF calculations. Within the BOA these electronic expectation values take into account the quantum character of the atomic nuclei. We have adopted the GAMESS program as an electronic structure tool [28]. As emphasized above the basis set adopted is of 3-21 G quality.

At the end of this section we summarize the approximations of the present theoretical approach. (i) We are working within the BOA, i.e. electron- phonon coupling terms in the electronic Hamiltonian are neglected. (ii) Under the restriction of the actual computational effort the empirical potential V(R) can only be an approximation to the potential energy surface of the ab initio HF formalism. In order to observe quantitative agreement between both poten- tials it is necessary to extend the parametrization of V(R) from the equilibrium geometry to the fall ab initio potential energy surface. (iii) It would be possible, at least in principle, to enhance the accu- racy of the ab initio part by extending the basis set and by going beyond the single-determinantal picture of the HF approximation. In our first step towards an all-quantum ab initio description of molecular sys- tems we wish to show a principal effect, i.e. the influence of the quantum character of the nuclei on expectation values of the electronic Hamiltonian.

380 R. Ramirez et al. / Chemical Physics Letters 275 (1997) 377-385

With this intention the limitations just mentioned should be acceptable. Our previous computational experience has shown that the statistical errors in the FPI simulations are small [13,29]. They are without influence on the results discussed in the next section. The FPI simulations have been performed on work- stations of DEC ALPHA type while the ab initio calculations have been done on a Pentium PC. The whole computational approach required several weeks of CPU time.

3 . R e s u l t s a n d d i s c u s s i o n

Let us start with a discussion of the FPI results to emphasize the importance of quantum fluctuations of the C6H 6 nuclei. At T = 0 K the here-called quan- tum delocalization dq of the nuclei defines the am- plitudes of the zero-point vibrations; see below. For definitions of the atomic delocalization parameters we refer to our recent work [12,13]. In Fig. 1 we

T:300'K.. ' 2 - -

=. A C-H

4- T= 50K

z ]i!! cross. 1 0 l J " l J I 8

T= 5K l

2 , .I, , I . 0 , T : 50K

80 100 120 140 160 160 ~ r[pm]

Fig. I. Radial distribution function (rdf) in arbitrary units o f the C - C and C - H bonds o f C r H 6. In the bottom diagram the results of a quantum simulation at T = 50 K have been portrayed. The upper three panels correspond to classical simulations. Note the different intensity scales in the diagrams.

T:3(30K C

~ class. 5

/, r T: 50K

2( cross.

0 I T: 5K

class.

I I T: 50K

. . . . . C-E-H g

A I I

-0.8 -0.6 -0.4 -0.2 " cos long[el

Fig. 2. Angular distribution function (add in arbitrary units of the C-C-C and C-C-H angles of the benzene molecule. The bottom panel is based on a quantum simulation (T = 50 K) while the three upper panels correspond to classical simulations. Note the different intensity scales in the diagrams.

have portrayed the radial distribution function (rdf) of both types of bonds of benzene derived by classi- cal simulations at different temperatures or by a quantum simulation at T = 50 K. In Fig. 2 we have plotted classical and quantum angular distribution functions (adD. The shortcomings of the classical approximation are clearly seen. In the classical limit there is no overlap between the rdfs of the C - C and C - H bonds even at RT. Both curves touch each other in the quantum simulation at T = 50 K. This computational result indicates that the molecular vi- brations of C6H 6 generate nuclear configurations with C - H bonds that approach the length of the C - C bonds. We wish to re-emphasize that both quantum distribution functions (rdf, ado are almost constant in the T interval between 0 and 200 K. This result implies that the thermal properties of benzene in this T-window are mainly determined by zero-point vi- brations. Even at R T only smaller shifts in the energy of the vibrational problem are detected. Strong deviations between classical simulations and the T =

R. Ramlrez et al. / Chemical Physics Letters 275 (1997) 377-385 381

50 K quantum simulation are found in the adf plots of Fig. 2 as well. Both figures demonstrate the importance of quantum effects even at room temper- ature. Fig. 2 indicates that the adf of the C - C - H angle is broader than the adf of the C - C - C angle (quantum simulation). This difference is strongly suppressed in the classical simulations. Such a level- ling of two quantities in the classical l imit is also found in the rdf plot of Fig. 1. It can be seen that the width of the two rdfs does not differ sizeably in the classical limit. The broader rdf of the C - H bonds with the light hydrogen atoms relative to the C - C bonds with two heavy atoms is a typical quantum effect. At T = 50 K the potential function V ( R )

adopted in the FPI simulations causes a quantum delocalization dq of C and H of 9.7 and 21.1 pm. As mentioned above these numbers approximate the am- plitudes of the zero-point vibrations; the thermal contributions to the atomic delocalization are small. At 750 K the overall spatial fluctuation of the H

atoms dto t = ~ + dt 2 (with d t denoting the classi-

cal thermal delocalization) is enhanced to 23.1 pm. The high-temperature value of dto t of the C atoms amounts to 14.3 pm.

In Figs. 3 and 4 we have studied the temperature influence on the rdf and adf in the quantum limit In addition to the T = 50 K simulations considered in Figs. 1 and 2 we have performed 750 K simulations.

"~ 1.0

f 0.8

0.6 ,_ 0./, l 0.2

0.0, ~ 0.08 ~ 0.0~,

i 0.00 -0.0~. -0.08

80

i

C6H 6 T=50K ---'--C-H -"" , ~ / ~ C-C

,/ ' \

. . . . . . .

100 120 It~0 160 180 - - r[pm]

Fig. 3. Top: Radial distribution functions of C 6 H 6 at T = 50 K. Bottom: Difference curve of the radial distribution function Ardf(T)= rdf(750 K)-rdf(50 K). Both diagrams refer to quan- tum simulations. Note the different intensity scales in the two representations.

10 == .~ 8 a 6

= L, 2 l0

1.0 .~ 0.5 • ~ o.o

l -O.S -1.0

-I.0

, , i i ! , , , ,

f.6H6 T=50K . . . . . . f-C-H

I I ¢ "

-0.8 -0.6 -0.4 -0.2 4 0.0

cos [angle]

Fig. 4. Top: Angular distribution functions of C 6 H 6 at T = 50 K. Bottom: Difference curve of the angular distribution functions Aadf(T) = adf(750 K) - adf(50 K). Both diagrams refer to quan- tum simulations. Note the different intensity scales in the two representations.

For convenience we have added the T = 50 K curves to Figs. 3 and 4. To provide an intelligible represen- tation in both figures we have plotted the difference curves Ardf(T) and Aadf(T) of both quantities (bot- tom diagrams). The Ardf(T) plot in Fig. 3 indicates that the influence of the temperature on the rdf of the C - C bonds exceeds the influence on the rdf of the C - H bonds. The rdf of the C - H bonds is a weak function of the temperature only. A strong broaden- ing of the rdf of the C - C bonds with increasing temperature can be extracted from the Ardf(T) curve. The adf results in Fig. 4 deserve some interest. The broader C - C - H adf in comparison with the C - C - C distribution at T = 50 K has been mentioned already in connection with Fig. 2. The Aadf curves, how- ever, demonstrate that the enhancement of T has a stronger influence on the C - C - C angular distribu- tion. This behaviour is a consequence of the heavier masses in the C - C - C fragment which cause an enhancement of the classical spatial degrees of free- dom of the nuclei.

To put the discussion of the electronic ab initio HF results onto a general basis we would like to mention the physical origin of covalent chemical bonding. For comprehensive discussions of this topic we refer to the interesting articles in Refs. [30-32]. Covalent bonding is driven by a reduction of the

382 R. Ramlrez et al. / Chemical Physics Letters 275 (1997) 377 -385

electronic kinetic energy into the direction of the bonds. The underlying mechanism is as follows. The interatomic sharing in the direction of the bonds causes an enhanced electronic spatial uncertainty. According to the Heisenberg uncertainty principle a reduced momentum uncertainty occurs. Smaller mo- menta become of higher probability. In the literature it has been emphasized in detail [32] that it is not the electron-core attraction which is responsible for co- valent bonding. On the contrary, this interaction is retarding. Note that the valence electrons have been transferred from the atomic region with a steep attractive potential into a region with a flattened potential gradient in the transition from the separated atoms to the molecule.

In Table 1 we summarize the results of an ab initio HF energy fragmentation of C 6 H 6 derived under different computational conditions. The first column of numbers E i corresponds to a single calcu- lation for the geometry optimized with the potential V(R) . In addition to these "single-configuration" results we have given "ensemble averages" E i sam- pled over 6000 configurations of C 6 H 6 which are populated in thermal equilibrium. From these num- bers it can be extracted that the contributions of the quantum fluctuations of the nuclei to the ensemble averages (i.e. energy increments at T = 50 K relative to the single-configuration results) exceed the ther- mal contributions (i.e. energy differences between the 50 and 750 K simulations). Remember that the

Metropolis technique adopted in the FPI approach generates the correct statistical weights of the corre- sponding C 6 H 6 geometries within the canonical en- semble.

The data in the Table visualize sizeable redistribu- tions between the kinetic and potential energy part of the C r H 6 Hamiltonian as response to the atomic spatial uncertainty. The electronic kinetic energy E~n calculated for the optimi__zed C 6 H 6 geometry exceeds the ensemble average Eki n at T = 50 K by 0.1513 E h. Remember that this shift is caused mainly by zero-point vibrations. Enhancement of T from 50 to 750 K is accompanied by an additional reduction in ffSk~ n of about 0.0328 E h only. The reduction in ff~kin due to the atomic fluctuations is, however, overcom- pensated by_a rise in the potential energy ffSpo t. As a net effect Eto t (ffSto t = ffSki n + ffTpo t) of the electronic Hamiltonian is destabilized under the influence of the nuclear dynamics. At T = 50 K the ensemble average Etot is 0.0573 E h above the single-config- uration value of Eto t. This difference is enlarged to 0.0779 E h when sampling over the high-temperature nuclear distribution. The changes in Eki ~ a n d Epo t under the influence of the atomic fluctuations have a non-vanishing influence on the virial coefficient f = EpoJEki n of the ab initio Hamiltonian. The ensemble averaging is accompanied by a slight enhancement

of f = ff~pot/ff~kin. In the following we consider the different contri-

butions to Epo t to explain the electronic origin of the

Table 1 Energy fragmentation of C6H 6 according to ab initio HF calculations in a 3-21 G basis set. Eki n denotes the electronic kinetic energy, En, c the repulsion between the atomic cores, Ee~ the two-electron repulsion, Ee~,,,c the e lect ron-core attraction and Epo t the potential energy of

the electronic Hamiltonian. Epo t = E.u c + Eel + Eel,.,c. With the exception of the nuclear kinetic energy Eto t is the total energy of the C r H 6 molecule. Eto t = Eki n + Epo t. f abbreviates the virial coefficient of the electronic Hamiltonian. The first set of numbers E i ( i = kin, nuc, el, el,nuc, pot, tot) corresponds to a single calculation [geometry optimized by V(R)]. The second and third sets of numbers if2 i have been derived as ensemble averages over 6000 nuclear configurations (T = 50 and 750 K). For the two sets of averaged quantities we have given the standard deviation o-. All energy elements are given in E h

Quantity Geometry optimization via V (R ) T = 5 0 K T = 7 5 0 K

i E i El Ei

i = kin 228.471056 228.319783 (0.3492) i = n u c 201.715841 200.707068 (2.1436) i = el 276.755518 275.708596 (2.0950) i = e l , n u c - 9 3 6 . 3 5 7 0 1 6 - 9 3 4 . 0 9 2 7 8 9 (4.5948) / = p o t - 4 5 7 . 8 8 5 6 5 7 -457 .677125 (0.3562) / = t o t - 229.414601 - 229.357342 (0.0168) f - 2.00418 - 2.00455 (0.0015)

228.286494 (0.4227) 200.427139 (2.8097) 275.424529 (2.7481) - 933.477820 (5.9885) - 457.623153 (0.4307) - 229.336658 (0.0220) - 2.00461 (0.0018)

R. Ramfrez et al. / Chemical Physics Letters 275 (1997) 377-385 383

enhancement of the ensemble averaged ffSto t. The spatial fluctuations of the nuclei are accompanied by a strong reduction in the electron-electron repulsion Eel. The ensemble averages at T = 50 and 750 K differ by roughly 1.05 and 1.33 E h from the single- configuration value. Remember that we discuss en- ergy increments larger than 28.5 and 36.1 eV. The large energy value at T = 50 K again indicates the dominant influence of the C 6 H 6 quantum fluctua- tions. Thermal effects are less pronounced. The col- lection of the ab initio HF results indicates that it is not only the electron-electron repulsion which is attenuated due to the spatial fluctuations of the nu- clei. The reduction of the electrostatic repulsion of the atomic cores is of the same order of magnitude. To sum up, we have three terms of the ab initio Hamiltonian which are reduced as a response to the nuclear spatial degrees of freedom. These elements measure the energetic costs encountered in the elec- tronic Hamiltonian, i.e. the two-electron and core- core repulsion as well as the electronic kinetic en- ergy. However, even the combined effect of these elements is overcompensated by the electron-core attraction, whose contribution to the electronic Hamiltonian is an obstacle for the formation of covalent bonding. Eel .... is destabilized under the influence of the nuclear dynamics. At T = 50 K this shift amounts to 2.26 E h, at T = 7 5 0 K it is en- hanced to 2.88 E h. The energy fragmentation in Table 1 shows an interesting effect. The ensemble averaging over 6000 configurations yields a net destabilization of Eto t of an order of some hun- dredths of a E h. However, the changes encountered in the kinetic and potential energy part of the Hamil- tonian are large. The small net effect ( = small in comparison with the changes in ff2ka n, ffJnuc, ff~el, Eel .... ) in ffSto t is the result of a strong competition between large numbers. The terms contributing to ffSto t are shifted into different directions. The broad distributions of certain energy increments in the Table can be extracted from the calculated standard devia- tion o- of the corresponding quantities sampled over 6000 benzene configurations. The largest o- num- bers occur in the electron-core attraction as well as in the nuclear-nuclear and two-electron repulsions. o- is reduced by one order of magnitude in the distribution of Epo t. Note the similarity in the o- values of Epo t and Eki n. The aforementioned compe-

tition of the different terms of Eto t leads to an additional suppression in the corresponding standard deviation o-. The width at half-maximum A E in the distribution of Eki n a m o u n t s to roughly 0.9 E h. It is enhanced to almost 5 E h for Eel and Enu c. This completes, to the best of our knowledge, the first microscopical explanation of the factors which lead

to a raise in the molecular potential e n e r g y ff~tot under the influence of the nuclear dynamics.

In Fig. 5 we have portrayed the distribution func- tion of Eto t at T = 50 and 750 K. Both curves are unsymmetrical. The asymmetry, however, is strongly enhanced in the high-temperature profile. The gradi- ent into the direction of higher energies is smaller than the gradient of the ascent branch. An increase in T leads to a pronounced broadening in the distribu- tion function. The width at half-maximum of the 50 K curve amounts to roughly 0.04 E h while one observes 0.05 E h for the high-temperature run. The asymmetry of the two curves in Fig. 5 i m p l i e s that the ensemble averages of Eto t, i.e. the Eto t values in Table 1, differ from Eto t a t the maximum of the distribution curves. The physical implication that follows from the numbers in Table 1 should be self-explanatory. Electronic expectation values mea-

D -

n i n / s0K JX ,

°,n

/ '\ / / l , , 2 "2 -22g.t+5 -229.6,0 -229.35 -22~30 -229.25 -229.20

Etot [au]

Fig. 5. Distribution function (arbitrary units) of the ab initio total energy Eto t of CrH 6 evaluated on the basis of 6000 different nuclear configurations generated by FPI simulations at T = 50 and 750 K. The basis set employed is of 3-21 G quality. Eto t is given in E h. The dotted vertical line symbolizes Eto t derived for the single-configuration calculation (min) and the broken lines denote the ensemble average Etot at T = 50 and 750 K (from left to right).

384 R. Ramfrez et aL / Chemical Physics Letters 275 (1997) 377-385

sured experimentally correspond to ensemble aver- ages as discussed in the present work. The corre- sponding quantities cover the superposition of all nuclear configurations populated as response to the quantum character of the atomic nuclei. We assume that the difficulties of quantum chemical methods in reproducing Compton profiles must be traced back to these atomic fluctuations leading to a large differ- ence between Eki n derived for the energy minimum and the ensemble average Ekj,-

At the end of this section we consider a model parameter which measures the deviations from the benzene D6h symmetry in the nuclear configurations populated in the canonical ensemble, i.e. the magni- tude of the electric dipole moment I/x] averaged over the 6000 nuclear configurations I~1. This quantity is important for any intermolecular interaction under the participation of induced dipole moments. In anal- ogy to the zero dipole moment of benzene in a frozen D6h geometry the ensemble averaged dipole moment ~, of benzene is zero. In fact we have observed some small deviations from ~ = 0 D in the third decimal digit. In contrast to the dipole moment the T-dependent ensemble averaging for [/xl yields a number 4= 0. At 50 K I~1 amounts to 0.277 D. At 750 K this number is enhanced to 0.350 D. Ensem- ble averaging of the magnitude of the cartesian components I~il (i = x, y, z) of the electric dipole moment /x shows a high anisotropy in the low-tem- perature simulation but an almost perfect isotropy in the high-temperature case.

4. Resum~

In this work we have derived electronic expecta- tion values of C 6 H 6 as ab initio HF based ensemble averages over 6000 nuclear configurations populated in thermal equilibrium. The corresponding configura- tions have been generated by Feynman PI simula- tions. Within the Born-Oppenheimer approximation we have taken into account the quantum character of the atomic nuclei when calculating electronic expec- tation values. This all-quantum description of a molecule leads to an intrinsic dependence of elec- tronic quantities on the nuclear spatial degrees of freedom even in the absence of an electron-phonon coupling term in the Hamiltonian. It has been men- tioned above that this ensemble averaging over nu-

clear configurations accessible in thermal equilib- rium is part of any experimental measurement of electronic expectation values for time windows larger than the frequency of molecular modes.

The fragmentation of the here-called total ab ini- tio energy Eto t and Eto t ( = potential energy of C6H 6

in the framework of the Born-Oppenheimer approxi- mation; among other elements Eto t contains the ki- netic energy of the electrons) has shown that the spatial delocalization of the nuclei is accompanied by sizeable redistributions between the electronic potential and kinetic energy. In the ensemble average Eto t one has a strong competition between large energetic shifts that act in opposite directions. It is the reduction in the electron-core attraction ffJe~,nuc which leads to a destabilization of the vibrating molecule relative to the molecule at rest. The FPI simulations of C 6 H 6 have shown the importance of the quantum character of the atomic nuclei for the evaluation of electronic quantities even at RT. The discussion of the rdfs and adfs has visualized the larger importance of quantum effects for the light hydrogen atoms in comparison with the C atoms.

Acknowledgements

This work has been supported by CICYT (RR) under contract No. PB 93-1254 and by the Fonds der Chemischen Industrie (MCB). We are grateful to G. Wolf for the preparation of the figures, to A. Gruss for her help in the generation of some ab initio data, and to S. Philipp, U. Schmitt and A. Staib for critically reading the manuscript and useful com- ments.

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