Steven A. Manson et al- The molecular potential energy surface and vibrational energy levels of methyl fluoride. Part II

8/3/2019 Steven A. Manson et al- The molecular potential energy surface and vibrational energy levels of methyl fluoride. P

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The molecular potential energy surface and vibrational energy levels ofmethyl fluoride. Part IIw

Steven A. Manson, Mark M. Law,* Ian A. Atkinson and Grant A. Thomson

Received 28th February 2006, Accepted 19th April 2006

First published as an Advance Article on the web 31st May 2006DOI: 10.1039/b603108k

New analytical bending and stretching, ground electronic state, potential energy surfaces for

CH3F are reported. The surfaces are expressed in bond-length, bond-angle internal coordinates.

The four-dimensional stretching surface is an accurate, least squares fit to over 2000

symmetrically unique ab initio points calculated at the CCSD(T) level. Similarly, the five-

dimensional bending surface is a fit to over 1200 symmetrically unique ab initio points. This is an

important first stage towards a full nine-dimensional potential energy surface for the prototype

CH3F molecule. Using these surfaces, highly excited stretching and (separately) bending

vibrational energy levels of CH3F are calculated variationally using a finite basis representation

method. The method uses the exact vibrational kinetic energy operator derived for XY3Z systems

by Manson and Law (preceding paper, Part I, Phys. Chem. Chem. Phys., 2006, 8, DOI: 10.1039/

b603106d). We use the full C3v symmetry and the computer codes are designed to use an arbitrary

potential energy function. Ultimately, these results will be used to design a compact basis for fully

coupled stretchbend calculations of the vibrational energy levels of the CH3F system.

1. Introduction

Methyl fluoride is a very important prototype for experimental

and theoretical studies in spectroscopy,110 structure and

bonding,1117 intramolecular dynamics3,4,6,1720 and reaction

dynamics.17,2123

The spectroscopy and dynamics of a five-atom molecule

such as methyl fluoride present a number of major challenges

to both theory and experiment. There are nine internal degrees

of freedom associated with the nuclear motion. Thus, themolecular potential energy function, which governs the vibra-

tionrotation dynamics, is a nine-dimensional (9D) hyper-

surface. Its full exploration using ab initio electronic structure

methods, including configurations probed near dissociation at

spectroscopic accuracy, is still an impossible task. For the

simplest five-atom system, methane, the best ab initio potential

energy surface is reliable up to only 13 000 cm1 above the

minimum energy configuration, which is one third of the

dissociation energy.24,25 Potential energy surfaces that do

represent dissociation for methane exist, but they combine

experimental and ab initio data.26

Similarly, accurate variational solutions of the 9D vibra-

tional Schro dinger equation for methane have been attempted

only very recently.24,25,2732 To our knowledge, only one

attempt has been made to calculate variationally and in full

dimension the vibrational energy levels of CH3F.2 However,

there were a number of severe restrictions on the method

described: the kinetic energy operator (KEO) was expressed in

normal coordinates, which are only suitable for studying small

amplitude vibrations; terms in the Hamiltonian operator were

neglected, the most serious of which was the restriction of the

potential energy to terms involving at most three normal

coordinates (this breaks the C3v symmetry and results in

degenerate levels being split by as much as 10 cm1 for

overtones of degenerate modes).

Most earlier ab initio studies of the potential energy surface

of methyl fluoride have taken one of two approaches: the first

involves exploring, at a modest level of theory, a region close

to the equilibrium configuration that can be described ade-

quately with a low-order Taylor expansion in normal mode or

other internal coordinates;2,11,13,15 the second approach ex-

plores, at a high level of theory, a few critical points on the

potential energy surface.14,16,17,21,22 A third approach, taken

by Luckhaus and Quack,4 was to focus on the three-dimen-

sional potential energy surface of the CH chromophore in

CHD2F. Here we will consider, using a relatively high level of

theory, two reduced subspaces of the methyl fluoride potential

energy surface (five-dimensional (5D) bending and four-

dimensional (4D) stretching) each including large distortions

from the equilibrium geometry. We consider determination of

these surfaces to be an important first step towards determin-

ing a full 9D ab initio potential energy surface for this

prototype molecule.

The choice of separating bending and stretching is moti-

vated by our intention to tackle the vibrational calculations

by solving initially separate bending and stretching vibra-

tional Schro dinger equations (before combining the eigen-

functions of the low-dimension Hamiltonians to form a

basis to tackle the full 9D problem). This approach has

already been taken for methane.2729,31 We note that this

may not be the optimum strategy since in the methyl

halides (and in methane) the CH stretching modes are very

strongly coupled to the bending modes by Fermi reso-

nances.1,3,6,19

Chemistry Department, University of Aberdeen, Meston Walk,Aberdeen, UK AB24 3UE. E-mail: [email protected] For Part I see ref. 41.

This journal is c the Owner Societies 2006 Phys. Chem. Chem. Phys., 2006, 8, 28552865 | 2855

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Knowledge of the molecular potential energy surface is a

prerequisite for determining vibrationrotation (and other)

properties. On the other hand, the ability to compute vibra-

tionrotation energy levels (and transition intensities) opens

up the possibility of extracting detailed information on the

potential energy surface from experimental high resolution

spectroscopic data.33 Previous efforts to derive potential en-

ergy surface information for methyl fluoride from spectro-

scopic data12,15 have relied on perturbation theory or other

approximate methods to compute ro-vibrational properties.

The application of perturbation theory to methyl fluoride (and

its isotopomers) is severely complicated by the presence of

many Fermi and Coriolis resonances.15

In this context, it is important to note that methyl fluoride is

an important prototype for developing and testing new ab

initio electronic structure methods.13,34 High resolution spec-

troscopic data afford the most stringent tests for comparison

with ro-vibrational properties computed from an ab initio

molecular potential energy surface.

The ability to calculate rotationally and vibrationally ex-

cited bound-state energies and wavefunctions precisely and

efficiently for molecules such as methyl fluoride would un-

doubtedly overcome the limitations currently imposed by the

enforced reliance on perturbation theory.

Ultimately, our goal is to develop a computational method

to solve the ro-vibrational Schro dinger equation for species

such as CH3F accurately and over a wide energy range. The

accomplishment of such methods for three-atom33,35,36 and

four-atom systems33,37,3840 have resolved difficult dynamical

problems, including the interpretation of high resolution mo-

lecular spectra, and have allowed the determination of accu-

rate potential energy surfaces (PESs) by fitting to such data.

The structure of the present paper is as follows. Details of

the determination of the ab initio potential energy at points on

extensive grids in stretching and bending subspaces are given

in Sections 2.1 and 2.2. The subsequent fitting of suitable

analytical functions for the stretching and bending surfaces are

reported in Sections 2.3 and 2.4 along with discussion of the

accuracy and range of the two surfaces. Our stretching and

bending vibrational energy level calculations are described in

Section 3. These calculations make use of the exact vibrational

kinetic energy operator derived in Part I.41

2. Potential energy surfaces

2.1. Computational method and basis set

All our ab initio calculations were undertaken at the CCSD(T)

level of theory using the MOLPRO package.42 We have used

the BornOppenheimer approximation and consider only the

ground electronic state. We have not attempted to make

relativistic nor other, small, corrections.43,44

We denote the basis set used (A)VT/QZ. It comprises the

(s,p,d,f) functions of Dunnings aug-cc-pVQZ basis for carbon

and fluorine, with the (s,p) functions of the aug-cc-pVTZ basis

plus the (d) function of the cc-pVTZ basis for hydrogen. This

gives a basis set consisting of 178 contracted Gaussian-type

orbitals in total. This basis set has been used successfully in an

investigation of five stationary points on the potential energy

surface for the CH3Cl + F- CH3F + Cl

, SN2 reaction.21

For the present purpose, calculating the potential energy at a

wide range of configurations of the CH3F molecule, this basis

set (and CCSD(T) level of theory) provides a good compro-

mise between accuracy and speed of calculation. A single point

energy calculation (at a C3v symmetry configuration) typically

took about 13 CPU minutes on a Compaq Alpha XP1000/667

workstation. Calculations at geometries distorted away from

C3v symmetry took up to four times longer.

Test calculations using the full aug-cc-pVQZ basis took over

five times longer than the corresponding calculations using the

(A)VT/QZ basis with only marginal gains in reproduction of

the experimental geometry. The aug-cc-pVQZ basis was not

considered further.

The optimised equilibrium geometry determined by the

CCSD(T)/(A)VT/QZ method is: RCHe = 1.0899 A , RCFe =

1.3872 A and bHCFe = 108.721. This agrees reasonably well

with the experimental geometry: RCHe = 1.0870 A , RCFe =

1.3827 A and bHCFe = 108.671.

14

The harmonic wavenumbers for 12CH3F (o1. . .o6) com-

puted using the CCSD(T)/(A)VT/QZ method are 3049.10,

1494.21, 1065.70, 3140.49, 1512.35, 1205.35 cm1. All of these

except o1 agree with the empirical values within the estimated

uncertainties associated with the latter.12 We note that the

wavenumber o1 is the most strongly affected by Fermi reso-

nance and other anharmonic corrections.12

2.2. Calculation of 4D stretching and 5D bending ab initio

grids

The computational demands of calculating multidimensional

ab initio PESs make it impractical to carry out the full

calculation on a single workstation. We have used the mas-

sively parallel CSAR supercomputer facilities at Manchester

University in order to determine the required ab initio energies.

The surfaces were initially determined by performing electro-

nic structure calculations at a large number of molecular

geometries to generate 4D stretching and 5D bending grids

of data points. CH3F possesses three equivalent hydrogen

atoms and so there are a number of geometries of the molecule

which, by symmetry, will all have the same energy. Thus we

calculated only symmetrically unique points.

The geometry of the molecule was specified using four

internal bond vectors (three CH and one CF) whose relative

orientation was defined using the polyspherical coordinates b

and f as described in Part I.41 Briefly, the radial coordinates

t1, t2, t3 and t4 correspond to the three CH bond lengths and

the CF bond length, respectively. The angle bi is the angle

between the CF vector and the CH vector for Hi (for i= 1,2

and 3). The angle f3 is the dihedral angle between the

H1CF and H2CF planes whilst f2 is the dihedral angle

between the H3CF and H1CF planes. We have calculated the

energy at over 2000 (symmetrically unique) stretching points

and 1200 (symmetrically unique) bending points. In choosing

the grids of points, the following constraints have been

employed:

(i) All points are less than 30 000 cm1 above the equili-

brium energy.

2856 | Phys. Chem. Chem. Phys., 2006, 8, 28552865 This journal is c the Owner Societies 2006


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For the stretching surface:

(ii) The b and f angles are held at their calculated equili-

brium values;

(iii) The four bond lengths are varied in the range 0.40 to+1.0 A from their equilibrium values;

(iv) Points are calculated at regular 0.1 A intervals in each

dimension, with the interval being reduced to 0.05 A close to

equilibrium.

For the bending surface:

(v) All four bond length vectors are held at their calculated

equilibrium values;

(vi) The HCF angles (b) are varied in the range 551 to 1701,

for each of the 33 unique combinations of interplanar angles

(f)the latter ranging from 501 to +501 from equilibrium;(vii) For the f angles, points are calculated at 101 intervals;

(viii) For the b angles, five points spaced at equal energy on

either side of the calculated equilibrium b value are used, [55.0,

60.0, 66.0, 74.0, 84.0, 136.0, 147.0, 156.0, 164.0, 170.0]1, giving

11 points in each b dimension.

In order to produce an analytical representation of the

potential energy surface for use in dynamical calculations, it

was necessary to fit the ab initio energies computed at the

points discussed above to suitable functional forms. Fitting

molecular potential energy hypersurfaces is a challenging

problem,19,45,46 but the powerful interactive non-linear least

squares fitting program I-NoLLS45 is available and has been

used to fit suitable functional forms to the ab initio grid points.

We have exploited the full C3v symmetry in the fits. The fitted

potential energy surfaces are available as FORTRAN sub-

routines from the corresponding author on request.

2.3. Stretching potential energy surface

The functional type selected to fit the stretching potential

energies has the following form

Vr1; r2; r3; r4 Xi;j;k;l

Ci;j;k;lfr1ifr2jfr3kfr4l 1

where i, j, k and l give the degree of the expansion in each

coordinate, with i, j and k being the three CH indices and l

the index for the CF mode and

f(rn) = 1 exp [an(rn rne)]. (2)In fitting the stretching PES, the Ci,j,k,land an parameters were

adjusted in the fit, with the rne being held at the values

corresponding to our calculated ab initio equilibrium geome-

try. Only symmetrically unique Ci,j,k,l

coefficients were in-

cluded in the fit. For example, in the most general case there

are six symmetrically equivalent coefficients that may be

obtained by permutation of the i, j and k indices, but only

one representative coefficient was varied independently.

The indices were varied to include all permutations for

which i + j + k + lr 6, with the C0000 parameter fixed at

zero (taking the equilibrium as the zero of energy). Any

adjustable parameters with a 95% confidence value greater

than that of the parameter itself were fixed to zero before

continuing with the fit. Points with energy 429000 cm1

above equilibrium were excluded because of the difficulty

reproducing such points accurately.

Thus, 53 symmetrically unique parameters were optimised

to fit the 2028 equally weighted data points, with an average

deviation of 1.8 cm1. The ab initio energies are fitted to within

a few cm1 except for above 20 000 cm1, where there are 12

points deviating by between 10 and 40 cm1.

The resulting surface has been examined carefully and

reflects the correct physical behaviour at short CH and

CF bond lengths. The surface also shows realistic behaviour

at large stretching geometries, although we do not expect it to

represent dissociation accurately.

2.4. Bending potential energy surface

A power series expansion was also used in fitting a functional

form to the bending ab initio potential energies. However, the

bending case is complicated by the presence of the angular

redundancy. In order to fully exploit the C3v symmetry during

the fit it was necessary to reintroduce the redundant angle into

the expression

Vb1; b2; b3;f1;f2;f3 Xi;j;k;l;m;n

Ci;j;k;l;m;nfb1ifb2jfb3k

ff1lff2mff3n3

where

f (bn) = (bn bne) (4)

and

f (fn) = (fn fne) (5)

The Ci,j,k,l,m,n parameters were floated in the fit, with bne and

fne being held at our calculated ab initio equilibrium geometry.

Inclusion of the redundant angle leads to complications in

the fitting procedure. For example, it can be shown that the

potential is always invariant with respect to terms of the type

Ca,a,a,b+1,b,b. Similarly, it can be shown that a number of terms

are not linearly independent, for example, C1,1,0,3,0,0 and

C1,1,0,2,1,0. In the case of these systematically correlated para-

meters, it was arbitrarily decided to remove the one with the

highest individual index before proceeding with the fit. As in

the stretching case, parameters with a 95% confidence limit

greater than the value of the parameter itself were also

removed before continuing with the fit.

The indices were chosen such that i + j + k r 10 and l +

m + n r 6, although only two parameters with i + j + k 4

8 were included. In addition, due to the large number of

possible parameters, the overall sum of the indices was re-

stricted such that i + j + k + l + m + n r 12. The C000000parameter was held fixed at zero.

Using 259 symmetrically unique adjustable parameters, the

1224 equally weighted data points were fitted with an average

deviation of 11.6 cm1. The bending PES is not fitted as well as

the stretching one. This is not unexpected: not only does the

extra dimension in the bending surface make it more challen-

ging but it also involves fitting two different types of coordi-

nate. The fact that a large number ofCi,j,k,l,m,n, parameters are

required to fit the surface to this level of accuracy indicates

that the functions used in the fit (eqn (4) and (5)) lack some of

the flexibility, and indeed basic shape, required to fit this type



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of surface to within a few cm1. It may be that the use of

cosine or Legendre functions would prove to be a better choice

for the b anglesthese functions having been used for similar

angular coordinates of smaller molecules.43 However, the

difference in performance from the function in eqn (4) was

not found to be significant in lower dimensional test cases. For

the f coordinates (unlike the stretching case), there appears no

obvious suitable functional form to use. This is a problem that

will require further investigation in order to fit the bending

surface with the same accuracy as the stretching one.

Although the bending PES is not fitted as accurately as the

stretching PES, it is still well within the accuracy of, for example,

the ammonia potential used in ref. 47. It is valid for use in a

variational calculation, up to the 30 000 cm1 region that the ab

initio points have been determined for. However, outside this

region there are areas of the surface that behave unphysically. In

particular, for areas of the potential corresponding to two or

more atoms approaching each other: intuitively, such regions

should have extremely large and positive energies. However, in

our fitted potential this is not the case. Consequently, if the

potential is to be used in dynamical calculations it is necessary to

take care in dealing with these unphysical regions. In the

variational calculations reported in Section 3.3 below, a scheme

of basis set contractions is used this has the effect of localising the

basis functions used in the final stage of the calculation, so as to

avoid the unphysical regions. We note that for 1D cuts through

the potential (where all other coordinates are held at their

equilibrium values) the surface does behave physically realisti-

cally at highly distorted geometries. Nevertheless, an important

piece of future work would be to consider re-fitting the bending

PES (and extending our angular grid) in an attempt to obtain a

more accurate fit and to also eliminate the unphysical regions.

3. Vibrational energy levels

3.1. Coordinate system

In order to simplify the KEO (see Part I41), we have used

coordinates based on orthogonal internal vectors. Three

Radau vectors are used to represent the CH3 subunit (this

allows the C3v symmetry of the molecule to be exploited). A

Jacobi vector then connects the F atom with the centre of mass

of the CH3 subunit. Our KEO is derived in terms of translation-

free internal vectors defined by a matrix V.41 The V matrix for

the present choice of orthogonal vectors is obtained from the

inverse of the M matrix of mass factors, which itself is formed

following the prescription of Schwenke48 as outlined below.

The Mmatrix is a 5 5 matrix of mass factors that gives theCartesian coordinates in terms of the internal coordinates (the

inverse of the relationship in eqn (6) of Part I41). The Jacobi and

Radau vectors are then constructed as follows. A Jacobi vector

connects the centre of mass of object A to object B, where A

and B are atoms or collections of atoms having masses mA and

mB (here A is the CH3 subunit and B the F atom). In this case,

the entries of the corresponding row ofMare mB/(mA + mB)for the atoms in object A and mA/(mA + mB) for the atoms in

object B.48 The three Radau vectors (numbered with an index k

= 1, 2, 3) describe the relative positions of the heavy central C

atom and the three light atoms Hj, j= 1,. . .,3. The entries of the

corresponding rows ofM are (g + e)mHj for the heavy atom and djk+ mH

je for the light atoms, where mC and mH

jare the masses of

the atoms. The total mass of the molecule is given by mT with g

= [1 (mT/mC)]1/2]/SjmHj and e = (gmC + 1)/mT.48Once the M matrix is constructed, it is straightforward to

obtain its inverse and hence the Vmatrix to be used in eqn (6)

of Part I41 to construct the internal coordinates. The Vmatrix

(obtained using the following masses (in atomic mass units):

mH = 1.007 825, mC = 12.0, mF = 18.998 403) that defines the

Radau/Jacobi orthogonal coordinate system used in this work

is (not to full precision and omitting the final column, which

locates the centre of mass)

V

0:964576 0:035424 0:035424 0:0670830:035 424 0:964576 0:035424 0:0670830:035424 0:035 424 0:964576 0:067083

0:000 000 0:000 000 0:000 000 1:0000000:893728 0:893728 0:893728 0:798750

0BBBB@

1CCCCA:

6The first three rows correspond to the hydrogen atoms, the

fourth to the fluorine atom and the last to the carbon atom.

Our internal coordinates are now defined as the four vector

lengths, r1,. . .,r4, three angles between the Jacobi vector and

each of the Radau vectors, b1,. . .,b3 and two angles between

planes, f2 and f3.

3.2. Stretching vibrational energy levels

The variational calculations of the stretching energy levels of

the C3v penta-atomic molecule CH3F make use of the vibra-

tional KEO derived in Part I41 and the stretching ab initio PES

described in Section 2.3. Although we discuss CH3F specifi-

cally below, our variational method is applicable to any

similar XY3Z system (including other methyl halides, and

species such as CH3D, CHD3, SiH3D and SiH3F).

3.2.1. Hamiltonian. The stretching KEO is expressed in

terms of the orthogonal coordinate system detailed in Section

3.1 above. For the stretching-only calculation, the bi and fjangles are held at their equilibrium values. From eqn (2) and

Table 1 of Part I,41 the stretching KEO is given by

Tstr h2

2

XN1i

V2aima

@2

@r2iXN1

i

2V2aimari

@

@ri

!7

where there is an implied summation over a = 1,. . .,N, where

N is the number of atoms.

Using this KEO and adding the potential energy, the matrixelements of the Hamiltonian are calculated asR

c*iHstrij cjr

21. . .r

2N1 dr1. . .drN1. (8)

However, as is commonly undertaken in order to facilitate the

evaluation of these integrals,4952 the radial part of the Jaco-

bian is incorporated into the operator and basis functions c as

follows:

c0 = r1r2r3r4c (9)

T0str r1r2r3r4 Tstr 1

r1r2r3r4

: 10



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The resultant form of the KEO is

T0str h

2

2

XN1i

V2aima

@2

@r2i; 11

with the same implied summation over a. Matrix elements are

now evaluated via the integral

Rc

0*

i H

0str

ij c

0

jdr1. . .

drN1. (12)

3.2.2. Basis functions, symmetry and kinetic energy matrix

elements. We have used Morse-oscillator-like basis functions

and evaluated the kinetic energy integrals analytically.53 We

have made no assumption about the form of the potential

energy and so it is necessary to evaluate the potential matrix

elements numerically, as described in Section 3.2.3.

A general 4D stretching basis function is denoted |abcxi,indicating x stretching quanta in the mode with vector

length r4 (nominally the CF mode here) and a, b, c

quanta in the three modes with vector lengths r1, r2 and r3(nominally CH modes here). In order to symmetrise these

functions, it is necessary to consider three distinct classes: Class

I, where the number of quanta in all three CH modes is the

same; Class II, where the number of quanta in two CH modes

is the same; and Class III, where the number of quanta in all

three CH modes is different. For the purpose of generating our

basis set, we use the convention a Z b Z c and thus obtain an

additional, but symmetrically identical, member of Class II.

The symmetries spanned by the three distinct classes are

Class I |aaaxi A1Class IIa |abbxi IIb |aabxi A1 + EClass III |abcxi A1 + A2 + 2E.

Symmetrisation of these basis functions can be achievedvia the promotion operator technique, employed on a number

of systems by Halonen and Child.54 A general symmetrised

basis function for the present C3v system is given by ref. 54

abcx;Gj i N1=2X

i

Ci ij i: 13

Where G is the symmetry species, N the normalisation con-

stant, Ci the expansion coefficient and |ii the basis member.The wavefunctions for each of the three symmetry classes are

given below.

| aaax,A1i

= | aaax

ijabbx; A1i 1ffiffiffi

3p jabbxi jbabxi jbbaxi

jabbx; Eai 1ffiffiffi6

p 2jabbxi jbabxi jbbaxi

jabbx; Ebi 1ffiffiffi2

p jbabxi jbbaxi

jabcx; A1i 1ffiffiffi6

p jabcxi jacbxi jbacxi

jcabxi jbcaxi jcbaxi

jabcx; A2 i 1ffiffiffi6

p jabcxi jacbxi jbacxi

jcabxi jbcaxi jcbaxi

jabcx; 1Eai 1

ffiffiffi1

p22jabcxi 2jacbxi jbacxi

jcabxi jbcaxi jcbaxijabcx; 1Ebi 1ffiffiffi

4p jbacxi jcabxi jbcaxi jcbax

jabcx; 2Eai 1ffiffiffi4

p jbacxi jcabxi jbcaxi jcbaxi

jabcx; 2Ebi 1ffiffiffi1

p22jabcxi 2jabcxi jbacxi

jcabxi jbcaxi jcbaxi14

Apart from the |abcx, 2Ebi function, which we find takes theopposite sign, our symmetrised functions agree with Halonen

and Child.54 This phase difference was not significant in the

work presented in ref. 54 but matters here because of our

method of computing the potential integrals, described in the

following section.

3.2.3 Potential matrix elements. The integration of the

potential matrix elements is carried out numerically in four

dimensions because of our assumption of an arbitrary form for

the stretching PES. We use the standard GaussLaguerre rela-

tion55 to evaluate the integrals as a sum of products of weights

(wi) and the value of the function f at M quadrature points (yi)Z10

eyyafydy XMi1

wifyi: 15

The points and weights are calculated using a modified version53

of the program given by Stroud and Secrest.55

Using eqn (15),

the potential integrals for the four dimensional stretching

problem have the form

habcx;GjVja0b0c0x0;Gi XM1

i

XM1j

XM1k

XM2l

wri; rj; rk; rl

cGabcxri; rj; rk; rlcGa0b0c0x0ri; rj; rk; rl

V

ri; rj; rk; rl

:

16

Here, (ri, rj, rk, rl) denotes a point in the 4D quadrature grid, w

(ri, rj, rk, rl) the corresponding product of weights and cG a

symmetrised wavefunction.

Calculating the potential matrix elements in this way is

computationally demanding and inefficient, as it involves

repeated evaluation of the potential at points related by

symmetry. For example, in the most general case where indices

i, j, k correspond to the three CH modes, there are six points

(corresponding to the permutations of the i, j, k indices) which

return the same value of the potential. Consequently, we make



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use of a slightly adapted form of the symmetrised quadrature

expression for XY4 molecules given by Xie and Tennyson27

habcx;GjVjja0b0c0x0;Gi XM1

i

Xij

Xjk

XM2l

w0ri; rj; rk; rl

V

ri; rj; rk; rl

Xha1

cGaabcxri; rj; rk; rlcGaa0b0c0x0

ri; rj; rk; rl:17

The quantity h is the degree of degeneracy and the new weight

is related to the unsymmetrised weight expression via

w0(ri, rj, rk, rl) = w (ri, rj, rk, rl)ch1, (18)

where c is the number of equivalent geometries (1, 3 or 6).

As a result of mixing the two components (Ga) of the

degenerate E representation, the expression for the sym-metrised quadrature is more complicated than eqn (16). The

use of this symmetrised quadrature formula reduces not only

the computational time required to perform the integration

but also reduces the memory requirement, because it is no

longer necessary to store the wavefunction at symmetrically

related geometries. The extent of the saving is dependent on

the problem under investigation, but the use of eqn (17) can

produce up to a six-fold saving in computational demand.

The basis set is characterised by the number N, which

represents the maximum order of the basis function in any

dimension. In addition, the constraint Sini r N is applied,

where ni is the number of quanta in the ith mode.

In the next two sections, CH3F stretching energy levelscalculated using our 4D ab initio stretching PES are reported.

3.2.4. Computational details. Morse-oscillator-like func-

tions may be variationally optimised for the system under

investigation by altering the Morse parameters oe, re and De.

The parameters oe, re and De are the harmonic frequency, the

equilibrium radial coordinate and the dissociation energy of

the corresponding Morse potential energy curve, respectively.

For the calculations on CH3F reported here, the optimised

Morse parameters are: oe = 2900 cm1, re = 1.130 A and

De = 35 000 cm1 for the CH modes; and oe = 1100 cm

1,

re = 1.431 A and De = 38 000 cm1 for the CF mode.

Our vibrational variational procedure uses the orthogonal

coordinates discussed above. However, our stretching poten-

tial is calculated and fitted in terms of bond length coordinates

(in each case the angular coordinates, b and f, are held fixed at

the appropriate equilibrium values). A problem arises in that it

is not possible to convert between the two coordinate systems

(either by refitting the PES or by converting each 4D ortho-

gonal grid point to the corresponding bond length, bond angle

grid point) in the reduced dimensionality stretching-only

problem. This is because converting between the two coordi-

nate systems also changes the angular coordinates. Conse-

quently, a full 9D bond length, bond angle PES would be

required in order to correctly carry out the conversion.

Of course, we could have calculated our 4D stretching and

5D bending CH3F potentials using the same orthogonal

coordinates used for vibrational energy level calculations.

However, as already discussed, the approach was to determine

reduced dimensionality PESs as an important first step to

obtaining an accurate, full 9D ab initio surface for CH3F.

Consequently, the PESs were determined at geometries defined

by bond length, bond angle coordinates because it was felt that

such coordinates would be more generally applicable than the

orthogonal coordinate system.

This does mean that a calculation using the variational

procedure described here and our present stretching PES is

not self-consistent and can not be readily made so. However,

we believe that the error introduced, especially for low lying

levels, is comparable with the difference that will arise between

the calculated and observed energy levels as a result of a lack

of stretchbend coupling. In fact, as has been similarly noted

in recent stretch- or bend-only calculations of methane,2729

the importance of stretchbend coupling in CH3F means there

is little to be gained from a straightforward comparison of a

large number of calculated and observed energy levels. How-

ever, comparison of at least a few calculated and observed

quantities (whilst remaining aware of the discrepancy caused

by the lack of stretchbend coupling) is still useful in that it

provides evidence that the variational code and the ab initio

PES produce physically realistic energy levels.

3.2.5. Results. Table 1 contains CH3F stretching energy

levels of all symmetries up to 10 000 cm1 above the ground

state. All levels given are converged to 0.1 cm1 or better. The

Table 1 Vibrational stretching energies for CH3F (in cm1) calcu-

lated using orthogonal coordinates. Excited state energies are givenrelative to the ground state

Symmetry Energy Symmetry Energy

Ground state A1 4997.8 I = 29 A1 7922.1I = 1 A1 1059.3 30 E 8051.3

2 A1 2105.5 31 A1 8074.03 E 2879.3 32 A1 8124.24 A1 2913.9 33 E 8346.65 A1 3138.7 34 A1 8346.96 E 3939.7 35 A2 8534.37 A1 3973.9 36 E 8550.88 A1 4159.2 37 A1 8588.19 E 4986.9 38 E 8589.4

10 A1 5020.4 39 A1 8695.611 A1 5167.1 40 A1 8811.112 A1 5666.3 41 E 8815.513 E 5671.1 42 E 8918.614 E 5774.7 43 A1 8955.315 A1 5814.2 44 E 9047.616 E 6021.1 45 A1 9051.217 A1 6053.3 46 A1 9101.218 A1 6163.0 47 E 9409.419 A1 6727.8 48 A1 9409.720 E 6732.5 49 A2 9597.021 E 6836.0 50 A1 9650.422 A1 6875.0 51 E 9651.623 E 7042.5 52 A1 9757.524 A1 7072.1 53 A1 9833.225 A1 7147.6 54 E 9837.426 A1 7776.0 55 E 9939.927 E 7780.6 56 A1 9972.828 E 7883.9



7/11

A1 symmetry calculations used N= 21 (giving a Hamiltonian

matrix of order 2678), N = 17 was used for the E levels

(matrix order 1974) and N= 15 for the A2 levels (matrix order

441). For the A2 and E symmetry types, convergence similar to

the A1 case is achieved with a smaller basis set as a result of

there being a lower density of states than in the A1 case. The

complete calculation took 14.4 h on a Compaq Alpha

XP1000/667 workstation.

Results are given in Table 2 for a calculation carried out

using the conventional bond length, bond angle approach.

This has been achieved by making slight modifications to the

stretching variational procedure already described. The major

change is the use of a stretching KEO derived in terms of bond

length radial vectors. This KEO may be readily obtained

because, as detailed in Part I,41 our approach to deriving the

operator was based on the use of a general set of radial

vectors. The only change to the KEO from that of eqn (11)

is the addition of terms involving second order derivative

operators coupling the radial vectors to one other.53

This change to the KEO and the use of a bond length V

matrix (see eqn (8) of Part I41) are the only alterations that

need be made to the computer code. This allows the stretching

ab initio PES to be tested in a variational calculation with the

same coordinates. It also gives an estimate of the error

introduced in our orthogonal coordinate calculations by using

a PES fitted in a different coordinate system. Fewer levels are

presented in Table 2 than for the orthogonal coordinate

calculation because the main purpose of the calculation is to

obtain an estimate of the stretchbend coupling, and this may

be achieved by comparison of only a few energy levels. All the

energy levels presented in Table 2 are converged to 0.1 cm1 or

better, the calculation is performed using a N = 11 basis for

the A1

and the E symmetry blocks. Note that the N= 11 basis

is sufficient to obtain the smaller number of levels in Table 2 to

the same degree of convergence (better than 0.1 cm1) as those

presented in Table 1. We compare the results calculated with

orthogonal and bond-length coordinates with a small number

of observables in Table 3.

The importance of stretchbend coupling in molecules such

as CH3F means that we would not expect the calculated and

observed results to be in particularly good agreement. Similar

observations have also been made in stretching-only calcula-

tions of methane.27 Additionally, for the stretching-only cal-

culation, the calculated results obtained using the different

coordinates systems would not be expected to agree.

Nevertheless, the results in Table 3 demonstrate that the

stretching variational codes and the CH3F ab initio stretching

PES produce physically realistic vibrational energy levels. The

pattern of levels and their symmetries are correct, while the

reproduction of the x33

anharmonicity constant is encouraging.

In order to investigate the anharmonic CH stretching part

of the potential, we have refitted our 1D CH ab initio points to

a Taylor expansion and used the resultant force constants to

obtain a value for the local mode CH anharmonicity, xCH(using second order perturbation theory56). The value of the

anharmonicity constant obtained is 62.5 cm1. This compareswell with the experimental value of Law6: 61.0 cm1. Inaddition, we have carried out some large A1 symmetry calcula-

tions in order to compare calculated and observed high energy

CH overtone transitions. The calculated values are obtained

using the bond length variational code described above. The

calculation uses a basis of NCH = 24 in the CH modes, with

NCF

= 21 for the CF mode (the total number of quanta in all

4 modes is r24), this results in a matrix of order 4218. Using

this basis, it has been possible to converge the results presented

in Table 4 to 1 cm1 or better. The results in the Table

demonstrate that for 36 quanta of excitation, the calculated

vibrational energy levels obtained are consistently too high. For

the higher observed states in CH3F, the motions are normally

considered as isolated CH stretching vibrations, nearly free of

the effects of stretchbend Fermi resonance coupling. The

overestimation of the vibrational energies together with the

value for the anharmonicity constant given above would sug-

gest that stretchbend coupling is having a significant effect on

Table 2 Vibrational stretching energies for CH3F (in cm1) calcu-

lated using bond-length coordinates and the N = 11 basis. Excitedstate energies are given relative to the ground state


Ground state A1 5208.4 I = 9 A1 5177.6I = 1 A1 1124.4 10 E 5269.0

2 A1 2234.1 11 A1 5475.93 A1 2941.4 12 A1 5825.5

4 E 3032.1 13 E 5877.25 A1 3329.2 14 A1 6003.06 A1 4066.9 15 E 6046.97 E 4158.0 16 A1 6273.68 A1 4409.8 17 E 6365.4

Table 3 Comparison of calculated results using orthogonal coordi-nates (Table 1) and bond-length coordinates (Table 2) with selectedobservable values (deperturbed for stretchbend Fermi resonance) forCH3F. All values are in cm

1

Assignment Observed Orthogonal calc. Bond length calc.

n1(A1) 2919.571 2913.90 2941.43

n3(A1) 1048.6164 1059.31 1124.44

n4(E) 2998.971 2879.32 3032.09

2n3(A1) 2081.3864 2105.48 2234.143n3(A1) 3098.44

64 3138.71 3329.212n4

2 (E) 6001.865 5671.08 5877.25x33

a 7.9264 6.57 7.37x33

b 7.9064 6.54 7.35a This value for x33 is calculated using n3 and 2n3.

b This value for x33is calculated using n3 and 3n3.

Table 4 Comparison of selected CH stretching results (calculatedusing the bond-length coordinates and NCH = 24, NCF = 21 basis)with corresponding observable values for CH3F; for each manifoldwith n quanta of total CH stretching excitation, the lowest A1 state isgiven. The n = 1 observed value is deperturbed for stretchbend Fermiresonances. All values are in cm1

No. CH quanta (n) Obs. Calc. (Obs. Calc.)/[n(n + 1)]1 29201 2941 10.52 58006 5826 4.33 85356 8620 7.14 11 1356 11288 7.65 13 6176 13834 7.26 15 97266 16266 7.0



8/11

the anharmonicity of the CH stretching energy levels. Con-

sidering the higher CH stretching vibrations as isolated Morse

oscillators, the effective CH stretching anharmonicity constant

contributes a term n (n + 1)xCH to these observed transitions.

Hence, column 4 of Table 4 shows the effective anharmonicity is

consistently too small in magnitude for the present calculations.

For the orthogonal coordinate calculation, it can be seen

from Table 3 that certain levels agree closely with the observed

values. However, the inconsistency in the use of coordinates

for this calculation may result in a fortunate cancellation of

errors. As with the bond length calculation, and as expected,

the discrepancies between the calculated and observed levels

become larger for the overtone levels. For the orthogonal

calculation, the error introduced by the inconsistency in the

use of coordinates makes further comparison of higher energy

overtone levels redundant.

3.3. Bending

The variational calculations of the bending energy levels of

CH3F use the vibrational KEO derived in Part I41 and the

bending ab initio PES described in Section 2.4. Again our

method is general for any similar XY3Z system.

The bending vibrational problem is more challenging than

the stretching one for several reasons. Firstly, there are five

bending degrees of freedom compared to just four in the

stretching case. Thus, the density of states is considerably higher

and the multidimensional quadrature used to evaluate potential

energy matrix elements is much more demanding. Also, depend-

ing upon the choice of angular coordinates, calculating the

bending energy levels of a centrally-connected penta-atomic

system may be further complicated by the presence of the

angular redundancy. This is seen in the work of Xie and

Tennyson28

and Mladenovic57

but our choice of coordinates

makes the redundancy relatively straightforward to deal with.

3.3.1. Hamiltonian and basis functions. For the bending-

only calculation, the radial coordinates are held at their

equilibrium values. The bending KEO is constructed using

eqn (2) of Part I41 and by selecting all gij and hi terms, in

coordinates b and f only, from Table 1 of Part I.41 This gives a

KEO consisting of twenty separate terms. Unlike the stretch-

ing case, no part of the Jacobian is incorporated into the

operator or basis functions. The bending Hamiltonian is

therefore integrated over the following volume element and

integration limits:

Zp0

sin b1db1Zp

0sin b2db2

Zp0

sin b3db3Z2p

0df2

Z2p0

df3

19The basis functions were chosen to be Legendre functions,

58

Pj (cos b), to describe the b motion and sin (kf) and

cos (kf) functions for the f motion. However, inspection of

Table 1 of Part 141 and eqn (19) shows that with this choice of

basis the cot2 and csc2 terms in the KEO give rise to singula-

rities with infinite integrals.

The most rigorous solution to this problem is to use instead

a coupled angular basis (associated Legendre functions).51

This results in a cancellation of these singular terms.

However, there are technical disadvantages associated with

the use of a coupled angular basis.51 For example, if a basis set

contraction approach is being used then a very large number

of eigenfunctions must be stored at each contraction stage.

Also, for XY3Z-type molecules, such as the methyl halides,

even at very high energies the singular points (b = 0, b = p)

will never be probed. Consequently, we can make use of the

direct product basis described above. Additionally, this leaves

open the possibility of a straightforward transformation to a

DVR representation, since the FBR approach will undoubt-

edly prove too computationally-demanding when applied to

the full 9D stretchbend vibrational problem.

Note that, although the singular points lie far from the

equilibrium geometry, the nature of the Legendre functions

means a quadrature based only on points far from either of the

integration limits (b = p, b = 0, the singular points) is

unlikely to be accurate. The difficulties associated with the

primitive Legendre functions are overcome by carrying out a

basis set contraction in each b coordinate. It is of course only

necessary to perform this for one b mode if all three are related

by symmetry. Using the primitive Legendre basis functions, we

diagonalise a 1D matrix in b, leaving all other coordinates

fixed at their equilibrium values. The coefficients of the

eigenvectors are then used to form the 1D contracted b

functions. For example, in b1

Fa1b1 XNP1j11

Pj1 cosb1Cj1a1 : 20

where NP1 is the number of primitive b functions, a1 runs

from 1 to the number of contracted functions used and Cj1a1 is

an eigenvector coefficient.

As b tends to zero (corresponding to the overlap of two

vectors) the potential becomes highly repulsive while as b tendsto p (corresponding to a linear XCH geometry in the methyl

halides) the potential becomes highly attractive. Consequently,

the new optimised functions rapidly approach zero at these

points and the integrals hFai|cot2bi|Faii, hFai|csc2bi|Faii maytherefore be evaluated accurately. The newly optimised func-

tions are also more physically suitable than the primitive

Legendre functions because they more closely resemble the true

eigenfunctions of the system. This reduces the size of the matrix

required to converge the full 5D bending calculation.

A 1D f contraction is also performed in order to obtain a set

of f functions that are more physically realistic than the

primitive sin (kf) and cos (kf) functions. The 1D f contraction

is carried out with b = p/2 for the KE contribution but at b = befor the potential integrals. This is necessary at this stage for two

reasons. Firstly, because of the singular terms in the KEO and

secondly, because isolated terms in the KEO are non-Hermitian

in our chosen basis. This approximation will not affect the

accuracy of our final result. Again, it is only necessary to perform

the contraction once as the f modes are related by symmetry.

The contracted f functions, for example in f2, are given by

Fb2f2 X1

k2MOP2sink2f2Ck2b2 ;

XMOP2k20

cosk2f2Ck2b2 ;

21



9/11

where MOP2 is the maximum order of primitive f functions, b2runs from 1 to the number of contracted 1D f functions used

and Ck2b2 is an eigenvector coefficient.

3.3.2. Symmetry and evaluation of matrix elements. Sym-

metrisation of the bending basis is more complicated than for

the stretching case discussed above. There are now two types

of coordinate and implications of the angular redundancy to

be considered. Due to the redundancy, we do not use the

coordinate f1. This means it is not possible, in our chosen

basis, to carry out a direct symmetrisation of the contracted

1D f functions in an analogous manner to the stretching case.

Instead it is necessary, as shown by Handy et al. in the case of

ammonia,59 to exploit the correlation between the C3v group

and its Cs subgroup.

The procedure involves carrying out a further 2D f basis set

contraction using the 1D f contracted functions already

obtained. The 2D f contraction is carried out in Cs symmetry,

producing A0 and A00 sets of eigenvalues and eigenfunctions. Itis noted that, as in the 1D contraction step, an approximate

form of the KEO must be used (that is, with bi = p/2) but

again this will have no effect on the accuracy of the final

answer. Inspection of the C3v/Cs correlation table60 shows how

the individual symmetry contracted functions may be isolated.

The eigenvalues and eigenvectors of the first contraction (A0)are stored, then after the second contraction (A00) the two setsof eigenvalues are compared. States that agree to within a

degeneracy threshold of 104 cm1 are identified as E type

contracted functions. The remaining A0 and A00 contractedfunctions may then be immediately identified as A1 and A2functions, respectively.

The full symmetrised 5D basis is then constructed by

combining these 2D f functions with symmetrised 3D b

functions (which are symmetrised in exactly the same way as

the three symmetrically equivalent stretching modes above)

using the appropriate vector coupling coefficients:61

A1 :A1bA1fA2bA2f

1ffiffiffi2

p EabEaf EbbEbf

A2 :A1bA1fA2bA2f

1ffiffiffi2

p EabEbf EbbEaf

Ea :A1bEafA2bEbfEabA1fEbbA2f

1

ffiffiffi2

p EabEaf EbbEbf

Eb :A1bEbfA2bEafEabA2fEbbA1f

1ffiffiffi2p EabEbf EbbEaf:

All the matrix elements for the full 5D bending calculation are

evaluated using these symmetrised 5D basis functions. For the

KE matrix elements, integration over the f modes is carried

out analytically,62 while for the b modes it is performed using

an appropriate Gaussian quadrature scheme.55 The factorisa-

bility of the KEO means these numerical integrals are inex-

pensive and can therefore be rapidly calculated to very high

accuracy. At each stage of contraction, the KE integrals are

stored for reuse in the full 5D calculation.

The integration of the potential matrix elements is carried

out numerically in five dimensions because of our assumption

of an arbitrary form for the bending PES. For the b modes, weuse a GaussLegendre quadrature scheme. Integration over

the f modes is performed using a trapezoid rule with quad-

rature points and weights

fk 2p

nfk 1

2

; wk 2p

nf; 22

where nf is the number of quadrature points and k = 1,. . .,nf.

The 5D quadrature formula is given by

ha1a2a3c23;G

jV

ja01a

02a

03c

023;G

i Xp

iX

p

jX

p

kX

q

lX

q

m

wbi; bj; bk;fl;fm

cGa1a2a3c23bi; bj; bk;fl;fm

cGa1a2a3c23bi; bj; bk;fl;fm

Vbi; bj; bk;fl;fm:23

Where (bi,bj,bk,fl,fm) denotes a point on the 5D quadrature

grid, w(bi,bj,bk,fl,fm) the corresponding product of weights

and cG a symmetrised wavefunction comprised of products of

1D b contracted functions (labelled by ai) a n d a 2 D f

contracted function (labelled by c23).

Calculating the potential energy matrix elements in this

manner is extremely computationally-demanding. Unfortu-

nately, with the present choice of angular coordinates and

quadrature points for the f modes, it is not possible to exploit

the full C3v symmetry molecular symmetry of an XY3Z

system. This is because not every point of our quadrature grid

is mapped onto the grid by the symmetry operations of the C3vgroup. However, this can be remedied63 within the framework

of our general approach. In the present work we have only

implemented a symmetry saving for the bending potential

integrals using the Cs group. This does not complicate the



10/11

expression given in eqn (23), as happened in the stretching

case, because the components of the degenerate representa-

tions are not mixed. Rather, eqn (23) is only slightly modified

such that the final summation runs to l rather than q.

The bending basis set is characterised using the notation:

{b,t,p}, where b gives the number of contracted b functions in

each mode, the constraint Si3bir t is applied and p gives the

number of contracted 2D f functions.

3.3.3. Results. Again, the bending-only vibrational calcu-

lations use coordinates based on the orthogonal vectors dis-

cussed above. The bending potential is, however, calculated

and fitted in terms of coordinates based on bond length

vectors. As above, a problem arises in that it is not possible

in the present work to convert between the two coordinate

systems, because this requires use of a full 9D PES. Again, at

least for low lying energy levels, the error introduced by this

inconsistency is comparable with the difference that will arise

between the calculated and observed energy levels as a result of

the lack of stretchbend coupling.

Table 5 contains CH3F bending energy levels of all symme-

tries up to just above 5000 cm1 above the ground state. All

levels given are converged to 0.3 cm1 or better. For A1 and A2symmetries, the calculation is carried out using a {9,12,21}

basis, producing an A1 matrix of order 803, and an A2 matrix

of order 716. The E symmetry calculation is carried out using a

{9,11,19} basis, producing a matrix of order 1036. The com-

plete calculation took three weeks on a Compaq Alpha

XP1000/667 workstation. The majority of this time was spent

performing the numerical integration of the potential energy

matrix elements.

3.3.4. Discussion. The importance of stretchbend cou-

pling in molecules such as CH3F means that we would not

expect the calculated and observed results to be in particularly

good agreement. Similar observations having also been made

in a recent bending-only calculation of methane.29 Neverthe-

less, the variational code developed in this work produces the

correct pattern of levels with the correct symmetries. The

results in Table 6 also demonstrate that the bending varia-

tional code and the CH3F ab initio bending surface produce

physically realistic vibrational energy levels.

The use of contracted basis functions as a way of overcoming

singularities in the KEO means that the bending code could not

be used to calculate all bound bending levels of a XY 3Z system.

However, the very high energy of the singular points means that

the code may be successfully used to calculate a very large

number of high energy bending vibrational levels.

4. Conclusion

We have calculated ab initio 5D bending and 4D stretching

potential energy surfaces for CH3F as an important first stage

towards a full 9D potential energy surface for this molecule.

Using these surfaces, we have calculated variationally highly

excited stretching and (separately) bending vibrational energy

levels of CH3F using a finite basis representation method. The

method uses the exact vibrational kinetic energy operator

derived in Part I41 and the full C3v symmetry is used to form

the final 4D and 5D basis functions. The computer codes are

designed to use an arbitrary potential energy function. Ulti-

mately, these results will be used to form a compact basis for

fully coupled stretchbend calculations of the vibrational

energy levels of the CH3F system.

If we retain the assumption of an arbitrary form for the

potential energy surface then the cost of numerical quadrature

will become prohibitive for full 9D vibrational energy level

calculations. Recent full-dimensional variational calculations

for methane have made use of the discrete variable representa-

tion (DVR) for some or all of the vibrational modes.31,32 In the

DVR basis, the potential energy matrix is diagonal so that

multidimensional quadrature over the internal coordinates is

not required. We have already computed accurate stretching

vibrational energy levels of CH3F using the DVR and poten-

tial-optimised DVR approaches (obtaining good agreements

with the FBR results reported above) and are currently

developing computer codes to tackle the full 9D vibrational

problem. This work will also include rotational motion.

Further consideration will be given to the choice of internal

coordinates. The latter will determine the degree of separation

of vibrational and rotational motions.33 The maximum

Table 5 Calculated vibrational bending energies for CH3F (in cm1).

Excited state energies are given relative to the ground state


A1(ground state) 3331.0 A2 4111.6E 1173.6 E 4167.0A1 1331.2 A1 4168.6E 1543.4 A2 4168.6E 2320.4 A1 4182.2A1 2328.2 A2 4183.4E 2517.6 E 4219.0A1 2673.8 A1 4241.4E 2683.5 A2 4244.3A1 2697.7 E 4259.0A2 2757.0 E 4408.1E 2885.7 A1 4420.5E 3056.1 E 4499.5A1 3067.0 E 4581.7E 3484.9 A1 4659.5A1 3598.1 E 4806.7A2 3608.5 E 4852.7E 3675.6 E 4967.9A1 3683.5 A1 4971.0E 3832.9 A2 4983.9E 3894.1 A1 5003.4E 4009.8 E 5042.9A1 4025.8 E 5065.1E 4037.3 A1 5070.6A1 4085.0 E 5196.9

Table 6 Comparison of calculated bending vibrational energy levelswith selected observable values (deperturbed for stretchbend Fermiresonance) for CH3F. All values are in cm

1

Assignment Observed Calculated

n2(A1) 1459.41 1331.2

n5(E) 1467.81 1543.4

n6(E) 1182.767 1173.6

2n2(A1) 2914.21 2697.7

2n50 (A1) 2921.7 1 3067.0n2 + n5(E) 2923.7

1 2885.72n5

2 (E) 2932.7 1 3056.1



11/11

separation is critical to successful solution of the full ro-

vibrational Schro dinger equation. The overall approach taken

here and in Part I41 will allow considerable freedom in the

ultimate choice of internal coordinates.

Acknowledgements

We thank the UK Engineering and Physical Sciences Research

Council (EPSRC) for access to the CSAR facility via the

ChemReact Consortium. We thank the Carnegie Trust for

the Universities of Scotland for supporting this work via the

awards of studentships to Steven Manson and Ian Atkinson.

We also thank a number of people for helpful discussions,

including Jonathan Tennyson, Junkai Xie, Mirjana Mladeno-

vic and Igor Kozin. Finally, we thank the referees for com-

ments on the manuscript.

References

1 M. Badaoui and J. P. Champion, J. Mol. Spectrosc., 1985, 109,402.

2 K. M. Dunn, J. E. Boggs and P. Pulay, J. Chem. Phys., 1987, 86,5088.

3 D. Luckhaus and M. Quack, Chem. Phys. Lett., 1992, 190, 581.4 T. K. Ha, D. Luckhaus and M. Quack, Chem. Phys. Lett., 1992,

190, 590.5 J. Lummila, L. Halonen, I. Merke and J. Demaison, J. Mol.

Spectrosc., 1996, 179, 125.6 M. M. Law, J. Chem. Phys., 1999, 111, 10021.7 D. Papousek, P. Pracna, M. Winnewisser, S. Klee and J. Demai-

son, J. Mol. Spectrosc., 1999, 196, 319.8 I. A. Atkinson and M. M. Law, J. Mol. Spectrosc., 2001, 206, 135.9 A. Baldacci, R. Visinoni and G. Nivellini, Mol. Phys., 2002, 100,

3577.10 A. Baldacci, R. Visinoni and G. Nivellini, Mol. Phys., 2004, 102,

1731.11 S. Kondo, Y. Koga and T. Nakanga, J. Chem. Phys., 1984, 81,

1951.

12 M. M. Law, J. L. Duncan and I. M. Mills, J. Mol. Struct., 1992,260, 323.

13 W. Schneider and W. Thiel, Chem. Phys., 1992, 159, 49.14 J. Demaison, J. Breidung, W. Thiel and D. Papousek, Struct.

Chem., 1999, 10, 129.15 I. A. Atkinson and M. M. Law, Spectrochim. Acta, Part A, 2002,

58, 873.16 G. F. Bauerfeldt and H. Lischka, J. Phys. Chem. A, 2004, 108,

3111.17 L. Wang, V. V. Kislov, A. M. Mebel, X. Yang and X. Wang,

Chem. Phys. Lett., 2005, 406, 60.18 D. Luckhaus, M. Quack and J. Stohner, Chem. Phys. Lett., 1993,

212, 434.19 M. M. Law and J. L. Duncan, Chem. Phys. Lett., 1993, 212, 172.20 R. P. A. Bettens, J. Am. Chem. Soc., 2003, 125, 584.21 P. Botschwina, M. Horn, S. Seeger and R. Oswald, Ber. Bunsen-

Ges. Phys. Chem., 1997, 101, 387.22 L. Sun, K. Song, W. L. Hase, M. Sena and J. M. Riveros, Int. J.Mass Spectrom., 2003, 227, 315.

23 J. M. Gonzales, W. D. Allen and H. F. Schaefer, J. Phys. Chem. A,2005, 109, 10613.

24 D. W. Schwenke and H. Partridge, Spectrochim. Acta, Part A,2001, 57, 887.

25 D. W. Schwenke, Spectrochim. Acta, Part A, 2002, 58, 849.26 R. Marquardt and M. Quack, J. Phys. Chem. A, 2004, 108, 3166.27 J. Xie and J. Tennyson, Mol. Phys., 2002, 100, 1615.28 J. Xie and J. Tennyson, Mol. Phys., 2002, 100, 1623.29 X.-G. Wang and T. Carrington, Jr, J. Chem. Phys., 2003, 118,

6946.

30 X.-G. Wang and T. Carrington, Jr, J. Chem. Phys., 2003, 119,94.

31 X.-G. Wang and T. Carrington, Jr, J. Chem. Phys., 2004, 121,2937.

32 H.-G. Yu, J. Chem. Phys., 2004, 121, 6334.33 J. Tennyson, Variational Calculations of Rotation-Vibration Spec-

tra, in Computational Molecular Spectroscopy, ed. P. Jensen andP. R. Bunker, Wiley, Chichester, 2000, pp. 305323.

34 S. Dressler and W. Thiel, Chem. Phys. Lett., 1997, 273, 71.35 S. Carter, I. M. Mills and N. C. Handy, J. Chem. Phys., 1993, 99,

4379.36 S. Carter and W. Meyer, J. Chem. Phys., 1994, 100, 2104.37 S. M. Colwell, S. Carter and N. C. Handy, Mol. Phys., 2003, 101,

523.38 S. Zou, J. M. Bowman and A. Brown, J. Chem. Phys., 2003, 118,

10012.39 I. N. Kozin, M. M. Law, J. Tennyson and J. M. Hutson, Comput.

Phys. Commun., 2004, 163, 117.40 I. N. Kozin, M. M. Law, J. Tennyson and J. M. Hutson, J. Chem.

Phys., 2005, 122, 064309.41 S. A. Manson and M. M. Law, Phys. Chem. Chem. Phys.DOI:

10.1039/b603106d.42 R. D. Amos, A. Bernhardsson, A. Berning, P. Celani, D. L.

Cooper, M. J. O. Deegan, A. J. Dobbyn, F. Eckert, C. Hampel,G. Hetzer, P. J. Knowles, T. Korona, R. Lindh, A. W. Lloyd, S. J.McNicholas, F. R. Manby, W. Meyer, M. E. Mura, A. Nicklass, P.

Palmieri, R. Pitzer, G. Rauhut, M. Schu tz, U. Schumann, H. Stoll,A. J. Stone, R. Tarroni, T. Thorsteinsson and H.-J. Werner,MOLPRO, a package of ab initio programs designed by H.-J.Werner and P. J. Knowles, Version 2002.1, 2002.

43 G. Tarczay, A. G. Csa sza r, O. L. Polyansky and J. Tennyson,J. Chem. Phys., 2001, 115, 1229.

44 O. L. Polyansky, A. G. Csa sza r, S. V. Shirin, N. F. Zobov, P.Barletta, J. Tennyson, D. W. Schwenke and P. J. Knowles, Science,2003, 299, 539.

45 M. M. Law and J. M. Hutson, Comput. Phys. Commun., 1997, 102,252.

46 R. Jaquet, Lect. Notes Chem., 1999, 71, 97.47 C. Le eonard, N. C. Handy, S. Carter and J. M. Bowmam, Spectro-

chim. Acta, Part A, 2002, 58, 825.48 D. W. Schwenke, J. Phys. Chem., 1996, 100, 2867.49 B. T. Sutcliffe and J. Tennyson, Mol. Phys., 1986, 58, 1053.50 N. C. Handy, Mol. Phys., 1987, 61, 207.

51 M. J. Bramley, W. H. Green and N. C. Handy, Mol. Phys., 1991,73, 1183.

52 T. J. Lukka, J. Chem. Phys., 1995, 102, 3945.53 J. Tennyson and B. T. Sutcliffe, J. Chem. Phys., 1982, 77,

4061.54 L. Halonen and M. S. Child, J. Chem. Phys., 1983, 79, 4355.55 A. H. Stroud and D. Secrest, Gaussian Quadrature Formulas,

Prentice-Hall, Englewood Cliffs, 1966.56 I. M. Mills, in MolecularSpectroscopy: Modern Research, ed. K.

Narahari Rao and C. W. Mathews, Academic Press, New York,1972.

57 M. Mladenovic, J. Chem. Phys., 2003, 119, 11513.58 G. B. Arfken and H. J. Weber, Mathematical Methods for Physi-

cists, 5th edn, Academic Press, 2001.59 N. C. Handy, S. Carter and S. M. Colwell, Mol. Phys., 1999, 96,

477.

60 E. B. Wilson, J. C. Decius and P. C. Cross, Molecular Vibrations,McGraw-Hill, New York, 1955.61 J. S. Griffith, The Theory of Transition Metal Ions, Cambridge

University Press, Cambridge, 1961.62 I. S. Gradshtein and I. M. Ryzhik, Tables of Integrals, Series and

Products, 4th edn, Academic Press, New York, 1980.63 X.-G. Wang and T. Carrington, Jr, J. Chem. Phys., 2004, 123,

154303.64 G. Graner, J. Phys. Chem., 1979, 83, 1491.65 E. W. Jones, R. J. L. Popplewell and H. W. Thompson, Proc. R.

Soc. London, Ser. A, 1966, 290, 490.66 J. S. Wong and C. B. Moore, J. Chem. Phys., 1982, 77, 603.67 E. Hirota, J. Mol. Spectrosc., 1978, 70, 469.


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