w.j. briels et al- hindered internal rotations in van der waals molecules and molecular crystals

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I N T E R N A T I O N A L J O U R N A L O F Q U A N T U M C H E M I S T R Y , VOL. XXIII . 1091-1 100 ( 1983 )

Hindered Internal Rotations in Van der WaalsMolecules and Molecular Crystals

W J . BRIELS, J. T E N N Y S O N , M . C L A E S S E N S , TH. V A N D E R L E E ,

A N D A . V A N D E R A V O I R D

Institute of Theoretical chemistry, University of Nijmeg en, Toernooiueld, Nijmegen, The Netherlands

Abstract

Th e quan tum dynam ical behavior of the Van der Waa ls molecule (N& and that of the ordered a and

y hases of solid N2 have recently been calculated, starting from the same ab initio Nz-Nz potential.

By interpre ting the results of these calculations we try t o improve our understanding of the librationlinternal

rotation motions of the N 2 monom ers and the orientational order-disorder (a-p)phase transition. Som e

new results a re presented and f urthe r (mean -field and libron-model) calc ulations are proposed which assess

explicitly the intermole cular pair correlation effects caused by th e anisotropic interaction potential.

1. Introduction

Van der W aals m olecules, which are w eakly bound complexes of normal molecules,

can be considered as small segments of molecular crystals. T he study of these com-plexes in conjunction with m olecular crystals is very illustrative and useful. In the first

place, their dynamical behaviour, although it lacks the collective aspects, shows many

of the typical chara cteristics found in molecular solids. He re, we concentr ate on or-

ientatio nal order-disorder transitions, i.e., the transitions from bending vibrations

(librations) to more or less free internal rotations. Classical Mon te C arlo calculations

[ 11 on ( C O Z ) ~lusters with n = 2 to 13 have shown that complexes as small as dimers

( n = 2) exhibit remark ably sh arp order-disorder phase transitions, as a function of

temperature. Since Van der Waals m olecules are considerably simpler than molecular

crystals, their dynamical properties can be calculated in much more detail by both

classical and q ua ntu m mechanical methods.

Second, Van der W aals m olecules provide the most sensitive tests of intermolecular

potentials, especially in the physically impo rtant region around the Van der W aals

minimum [2]. Although it has also been adv ocated [3] th at molecular crystals and ,

in particular, their phonon dispersion relations, ar e very useful in this respect, we note

tha t th e route from the intermolecular potential to the phonon frequencies that canbe compared w i t h experiment, involves approximations, such as the assumption of

pairwise additivity for the potential an d the (usually harmo nic) model for the lattice

dynam ics. For simple Van der Waals m olecules, on the other hand, accurate dynamical

calculations can be performed which lead directly from a given potential to the ro-vibrational spectrum.

As an example we study nitrogen. Apart from extensive work [3,4] on solid N z,

which has two ordered ( a nd y) hases and one orientationally disordered (0)hase,

there is also much interest in gas and liquid properties [5-71. An important part of

0 983 John W iley & Sons, Inc. CCC 0020-7608/83/031091- 10$02.00



1092 B R I E L S ET AL .

this work is concerned with extrac ting empirical N2-N2 interac tion potentials [3]

from the experimental da ta. Recently, a detailed N2-N2 potential has been obtainedfrom ab initio calculations [8,9]. Th is potential has been successfully used in lattice

dynamics studies [ 101 on a and y N2 and , at the same time, in accu rate dynamical

calculations [ l 11 on the Van der Waals molecule (N2)2. In the present paper, we

discuss some new results and ways to extract further useful information.

2. Lattice Dynamics of Solid N2

Using the ab initio N2-N2 potential in its site-site representation [9] with the force

centres shifted on the N-N axes, away from the nuclei, Luty et al. [101 have per-

formed lattice dynam ics calculations on the ordered cub ic ( a ) nd tetragonal (y) N2crystal phases. They applied not only the usual harm onic method [121, but also the

self-consistent phonon (SCP) method [121 which corrects for the anh armon icity of

th e potential by using effective force constants obtained by (qua ntum mechanical and

the rm al) averaging over the molecular displacements in the crystal. In t he general-ization of the SCP method from atom ic (ra re gas) crystals [ 131 to molecular solids

[141 i t had to be assumed, however, th at t he librations have small amplitudes.

Th e results [ 101 ar e generally in good agreement with experim ent: lattice constan ts,

cohesion energy, th e phonon frequencies and their pressure or volume dependence

(Griineisen parameters). This is the more satisfactory since no adjustments of the a b

initio potential to the crystal dat a have been made (in contrast with the usual semi-empirical treatments [3]). For the translational phonon modes the agreement is almost

perfect: th e frequencies are only 4% higher than the neutron scattering results (1 1%

higher if one neglects anharm onicity). For the librational modes the agreem ent is

somewhat less good: th e frequencies are overestimated by 28% on the average (34%

in the h armonic model). Also these modes do not soften sufficiently with increasing

temperature. This is not surprising, however, as the librational displacements are

actually not small even at low temp erature (estimated [3] root-mean-square dis-

placement 17 deg at T = 20 K). When the temperature rises to the a-/3 phase tran-

sition point, th e N2 rotations in the crystal become more and more free. So, one may

expect that the generalized SCP method [I4 1 breaks down for the librationalmodes.

A model which is better, in principle, a t describing the librations in solid N 2 and

their chang e into hindered rotations w hich occurs at the a-/3 phase transition is the

so-called libron model [15-1 71. In this model it is assumed that the angu lar vibrations

of the molecules can be expanded in a basis of free-rotor functions. In case of linear

molecules such as N2 these are simply spherical harmonics Y,,m (O ,@) , where 0 and

$I ar e the angular displacement coordinates which can adop t any value in the range

0 2 0 < ?r, 0 s @ < 27r. T h e lattice dynamics treatment star ts with a mean-fieldcalculation [17 ]; the wave functions emerging ar e then used in the “libron” modelwhich accounts for the pair correlation effects caused by the anisotropic intermolecular

potential, and , at the same time, introduces dispersion (dependence on the wave vector

q) into the libron frequencies.

For solid N2 several of such libron model calculations have been mad e [15-1 71,



VA N DEK W A A L S M O L E C U L E S 1033

always using approximate empirical potentials. The most advanced one is that by

Dunmore [16] who has summed the intermolecular potential over six shells (86

neighboring m olecules) in th e a - N z lattice, but has still neglected term s in the crystal

potential which have nonazimuthal symmetry around the equilibrium N2 axes (i.e.,

the 6 dependence of the mean-field potential). In all these libron treatments it has

been assumed that the translational vibrations of the molecules in the lattice are

completely decoupled from the angular motions. Actually, in crystals which have

inversion symmetry , such as N2, the translations and the librations do separate a t some

points in the Brillouin zone (for the optical modes at q = 0, for instance), but the

translational displacements still influence the effective librational potential. This effect

is neglected by keeping th e molecules with their centre s fixed at the lattice points.

Here, we present some new resu lts obtained by utilizing the same ab initio N2-N2potential [9] tha t was used previously in th e harmonic and SCP lattice dynamics cal-

culations [lo]. Instead of the site-site representation, we have now employed the

spherical expansion of this potential, however, which is more convenient in this case

and which has also been used for the (N2)z dimer [111 (see Sec. 3). Th e spherically

expanded potential reads for a pair of molecules A and B :

with R being th e distance between the centres of A and B , 8 A , @ A nd 0 B , @ B de-scribing the orientations of th e molecular axes and 0 and CP the orientation of the vector

R, relative to some lattice coordinate system . Th e expression in brackets is a 3- j symbol

[181. T he ab initio calculations [8,9] have demonstrated that anisotropic terms in

potential (1) up to LA = L B = 4 inclusive ar e importa nt; actually som e higher term swere included as well [9 ].

Sum ming this potential over six shells (86 neighbouring molecules), just as Dunmore

[161, but making no further approxim ations, mean-field calculations have been carriedout [191 for a-N 2 in the cubic Pa3 stru cture [4]. T he spherical harm onic basis was

extended t o j = 10 (66 functions) for ortho-N z and t o j = 9 (5 5 functions) for pa ra-N 2.

T he results listed in Tab le I dem onstrate that this is sufficient to converge the ground

sta te cohesion energy to within 0.5 cm-' and the first excitation energy to within 1

cm-' ; he mean -field iterations were always convergent in three cycles.

T he first excitation frequency 50 cm-I is very close to the value of 5 1 cm-I calcu-

lated by Dunmore [161, which is surprising as our anisotropic ab initio Nz-Nz po-

tential [9] is rather different from the empirical potential used by D unm ore [161, see

Table 11 . Th e difference between orf ho-N 2 and para -Nz crystals is small. Actually,

it would be interesting if this difference could be observed since i t presents a measure

for the degree of orientational rigidity of the N2 crystal phases (y, , an d p). In th e

com pletely rigid case, the ortho- an d pa ra -N z results would be identical [161; if th e

N2 molecules ar e free internal rotors, such as H2 molecules in the solid, the excitation



I094 B R I E L S ET A L

TABLE . Mean-field results for the a - N z crystal (lattice constant a = 5.644 A) sing

the ab initio N2-N2 potential [9] . Total librational energy Eo and first excitation

frequency w as functions of the spherical har mon ic basis size for orlho- and para-Nl.

o r t h o p a r a

2 ( 6 ) - 6 . 2 5 3 1 2 8 . 7 3 ( 1 0 ) - 6 . 5 5 1 100 .5

4 ( 1 5 ) - 6 . 7 1 1 7 9 . 0 5 ( 2 1 ) - 6 . 7 9 1 6 5 . 3

6 ( 2 8 ) - 6. 82 9 57.3 7 ( 3 6 ) - 6 . 8 4 4 5 3 . 0

8 ( 4 5 ) - 6 . 8 4 9 51 .2 9(55) - 6 . 850 5 0 . 4

l O ( 6 6 ) - 6 . 8 5 1 5 0 . 2

* lnclud cs the librational zero-po int energy (well depth E , = 7.436 kJ/mol), but not the

zero-point energy corresponding with the translational lattice vibrations. Experimental

cohesion energy 6.9 2 kJ/mol.

frequencies would differ by 8 cm-I. Also in the (N2)2 dimer there is a noticeable

difference between ortho-ort'ho, ortho-para, and para-para complexes [ 1 11.

It is striking that the mean-field spectrum is very similar to tha t of a two-dimensional

harmonic o scillator, just as Dunmore [ 161 has found w ith his empirical potential. This

justifies the application of his libron model, in order to take the pair correlation effectsinto account. It appeared [ 161 tha t, for q = 0, this gives a significant lowering of the

mean-field librational frequencies bringing them into good agreement with the ex-

perimental values. We intend to proceed with this model and, also, we will attempt

to include the effects of the translational lattice vibrations by combining the libron

model with the self-consistent phonon model or by a renormalization procedure.

3. Dynamics of the Van der W a d s Molecule (N&

T he questions which a re pertinent to the understanding of the librational m otions

in solid N 2 and the a-p phase transition, what is the importance of pair correlationeffects and how well does the libron m odel account for these effects, can be answered

explicitly for the (N2)2 dim er. Th e dynam ics of this dimer is also interesting by itself,

however, since th e (N2)2 complex can be expected to lie between a norm al molecule

with a near-rigid nuclear framework and a system in which the nuclear positions are

completely undefined. W e are familiar with the latter situation fo r electrons in atoms

and molecules; for the nuclei it is rather unusual, but it occurs in noble gas-H2 [2]

and (H2)2 [20,21] complexes where the H2 molecules are practically free quantum

mechanical in ternal rotors.

The ab initio potential [9] favors a crossed equilibrium structu re for the (N2)2 dimer

with D, = 122 cm-' and R e = 3.5 A. There a re different barriers to internal rotations:

25 cm-I through t he parallel stru ctu re with R, = 3.6 A and 40 cm-I through the

T-shaped structu re with R, = 4.2 A . Using the spherical expansion (1) of this po-

tential (in a body-fixed frame with 8 = = 0,$ = $ A - B ) , accu rate dynamical



VAN D E R WAALS MO L E C U L E S 1095

TABLE1 . Spherical expansion (1) of the ab inirio N2-Nz potential [9] , compared

with the empirical potential of Raich and Mills used by Dunmore [ 161. Th e expansion

coefficients U L ~ . L ~ , L ( R )re tabulated (in cm-I) for the nearest ( R = 3.991 A) nd next-nearest ( R = 5.644 &) neighbor distances i n the a - N z lattice.

+

I A LB

0 0 0

2 0 2

2 2 0

2 2 2

2 2 4

4 0 4

4 2 2

4 2 4

4 2 6

4 4 0

4 4 2

4 4 4

4 4 6

4 4 8

6 0 6

6 2 4

6 2 6

6 2 8

R = 3.991

e m p i r i c a l [ 1 6 1

-79.351

15.905

3.104

- 6.144

25.628

5.513

0.820

- 1.160

2.816

0.087

- 0.114

0.165

- 0 . 307

0.875

0.816

0.132

- 0 . 200

0.533

a b i n i t i o [91

-55.345

32.359

5 . 564

- 9.146

39.239

8 . 635

0.629

- 1.467

10.282

0 . 010

- 0.012

0.052

- 0.263

4.052

0.779

0,022

- 0.114

1.075

R = 5.644 1

e mpi r i c a l 1161

-17.834

- 1.311

- 0 . 053

- 0.112

2.833

0.035

0.001

- 0.002

0 . 010

0.001

0.002

0.001

ab i n i t i o [91

-13,831

- 1.011

- 0 . 015

- 0.002

2.555

0 . 007

- 0.001

0.356

0.089

0.001

0.013

calculations have just been finished [111. These start from the exact (rigid monomer)

ro-vibrational Ha milton ian including centrifugal distortions and C oriolis interactions

I221

where j A and j, ar e the monomer angu lar m omenta associated with the body fixed

angles (OA,4A) and (O&B) , respectively,j = j, + jB ; J describes the overall rotation;

p, p ~ ,nd p~ are th e dimer and m onomer reduced m asses; and r A and r B are the(fixed) N 2 monomer bond lengths. An a pprop riate and convenient basis to calculate

th e bound states of Ham iltonian (3) is the following:

~n ( R )vf:,s( 8 ~ 9 4 ~ 4 ~~ L , k ( a , P , o ) (4 )



1098 B R I E L S ET A L .

problem in electronic structure calculations. So the ro-vibrational states which emerge

from this calculation are fully correIated (within the given finite basis). Speaking interms of solid lattice dynamics, this means that all the (pair) correlations between the

monom er librations and also the coupling with th e translational vibrations (the dimer

stretch) a re exactly included. In analogy with electronic structu re theory, one can look

at the n atural orbitals which diagonalize the first order density matrix [25 ,26] ; his

density matrix can be obtained for each ro-vibrational state from the full configuration

interaction function in terms of basis (4) by integrating over the coordinates of all

particles except one. As particles one could take the individual nuclei, but it seems

more natural in this case where the centre of mass trans lation al motion is separated

off, the monomer stretch vibrations are decoupled and we are interested in th e

monom er librational m otions, the dim er stretch and the overall rotation to define thefollowing four “particles” (degrees of freedom): (0, ,4~),O B , ~ B ) ,R , (p,cu).T he

“n atu ral orbitals” ar e those orbitals for which th e “configuration interaction” ex-

pansion converges the fastest [ 2 5 ] .

In order to study th e effects of the pair correlations between th e librations and the

coupling with the dim er stretch vibration explicitly, we define the mean-field H am -

iltonian for the librations (for J = 0, where we have no Coriolis coupling with the

overall rotation)

H Y ~ = H A+ < $ B ( B B , ~ B ) I H A B I ~ B ( O B , ~ B ) ) , (7)

where the “one-particle” term an d the pair term are defined as

and th e mean-field wave functions \C /B can be expanded as

Since these functions C/B must be the eigenvectors of the mean-field Ham iltonian H F F

which is analogous to Eq. (7) and con tains the functions $ A , the problem has to be

solved self-consistently. Th e expectation values over the dimer stretch coordinate R

can be evaluated if this coordinate is also included in the mean-field procedure by

defining an effective radial Hamiltonian (averaging the full Hamiltonian ( 3 ) over

the angular wave functions \ C / A $ / B ) . Alternatively, one can decoup le the dimer stretch

motion by solving the m ean-field problem , Eqs. (7) to (lo), for fixed distance R. Doing

this for various R values, one obtains an effective radial potential in the Born-Op-

penheim er sense, which can be used, then, to ge nerate th e radial solutions.

T he latter way of decoupling the radial coord inate R from the angular p roblem by

freezing it at various values, has just been performed [ 2 7 ] ,not for th e mean-field

problem but fo r full H amiltonian (3) . The results for R = R, = 3.46 A and R = ( R )

= 3.79 8, are shown in Figure 2, together with the results of the full radial and angular

solution [ l I ] of Hamiltonian ( 3 ) . Th e comparison is quite interesting. A t R = R e ,all the an gular vibration frequencies are m uch too high and the order of the levels is



V A N D E R W A A L S M O L E C U L E S

crn-1 cm-1

I099

-L O

- 50

-60

-7 0

-80

2 1

1 2

2 0

0 1

1 1

1 0

0 1

0 0 -

In JE

Figure 2. Vibrational (J = 0) energy levels in (N2)z from the full dynamical [linear

combination of radial and angular momen tum products ( L C - R A M P ) ] calculation [ 1 11,

as compared with the angu lar vibrational levels calculated for fixed R (see text). Th e

free rotor ( R = a) evels are shifted (see right-hand energy scale) with respect to the

others.

incorrect. A t R = ( R ) , he ground sta te average distance in the full calculation, thelevels up t o ab out 20 cm-1 above the ground sta te ar e quite well represented. These

correspond with angu lar excitations [111; apparen tly, the barriers to internal rotations

felt by the monomers at R = ( R ) re similar to the effective barriers in the full cal-culation (whereas at R = R , they are much higher). Around 25 cm-l above the

ground state, the level ordering for R = ( R ) begins to deviate more strongly from theordering in th e full calculation. This is due to the interaction (Ferm i resonance) be-

tween the angular overtones and the first dimer stretch excitation at 33 cm-'. Weconclude tha t freezing R a t ( R ) which might be the exp erimental ground sta te av-

erage distance) yields reasonable values for the fundamental libration frequencies,

a s well a s for the tunneling frequency.

The energy levels calculated [27] for a range of (frozen) distances R form a se t of

potential curves for the radial problem. For the ground state the minimum in the curvelies a t R = 3.65 A, which is substantially larger tha n Re = 3.46 A . (Since the curve



1100 B R l E L S ET AL .

is asym me tric the av erage distanc e will still be larger; cf. ( R ) = 3.79 A in the full

calculation .) Solving for the levels in these one-dimensional effective radial problemsis called the Born-Oppenheimer angular radial separation (BOARS) method [ 281.

T h e shift in the lowest energy levels from the full ( L C - R A M P ) calculation to the R

= ( R ) evels in Figure 2 is due to the zero-point energy in the stretch coord inate R ,

which is not included in th e frozen R calculations. T h e results obtained for R = ( R )

indicate that the B O A R S method could give reasonable answers for the lowest excitation

frequencies, although th e conditions on which this schem e is based [28] ar e certainly

not fulfilled for (N2)2.

Ne xt, we wish to proceed with th e mean-field model, using th e proce dure ou tlined

in Eqs. (7) to ( lo) , and to test the various libron mod els [15-171 for describing the

effect of the pair correlations in (N2)2 and looking at orientational order-disordertransitions. T h e mean-field “orbitals” (10) can be useful, moreover, a s a basis for th e

full dynamica l problem, instead of the fr ee rotor basis ( 5 ) . They migh t help in reducing

th e size of the secular problem over the exact Ham iltonian ( 3 ) which becomes quite

formidable fo r the higher J states [ 11.

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