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On the Jahn-Teller distortion of CH4 +

R.N. Dixon aa School of Chemistry, The University, Bristol, BS8 1TS

Available online: 22 Aug 2006

To cite this article: R.N. Dixon (1971): On the Jahn-Teller distortion of CH4 + , Molecular Physics, 20:1,

113-126

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MOLECULAR PHYSICS, 1971, VOL. 20, No. 1, 113-126

On the Jahn-Teller distortion of C H 4 +

by R. N. D I X O N

School of Chemistry, T h e University, Bristol, BS8 1TS

(Received 20 April 1970)

Ab initio calculations of the vibronic potential surface of the 2T2 ground state of CH4 + have been carried out, using a contracted gaussian approximation to a double-zeta basis of Slater orbitals. The minimum energy is found for a tetragonal distortion to the point group D2n, with two HCH angles of 141 ~ and four of 96 ~ and all rcn equal. All calculations for distorted CH4 + were carried out with the mean rcH held at 1 "147 ~,, which minimises the energy for Ta symmetry. The distortion from Ta symmetry lowers the electronic energy by 1"41 ev. Distortions from Ta to Car lead to a maximum stabilization of 1"22 ev. Vibronic perturbation parameters derived from this surface are employed in discussions of the 2Tz-IA1 photoelectron transition of CHa and a possible 2E<-2B~ absorption transition of CH4 +. A vibrational interval of ~ 1200 cm -1 observed near the origin of the 2Tz--1A1 photoelectron transition is assigned to one component of v2 (calculated frequency, 1300 cm-1).

1. INTRODUCTION

T h e Jahn-Te l le r effect has been extensively invoked in the discussion of transition metal compounds [1]. However, there have until recently been few examples of Jahn-Tel le r interactions in orbitally degenerate states of simple molecules. Recent developments in photoelectron spectroscopy have indicated that the positive ions of many simple symmetrical molecules may be distorted in one or more electronic states as a result of vibronic interactions. Thus the photoelectron spectrum of methane with 21-21 ev photons consists of a broad band system with a width of ~ 3 ev and a complex envelope, the appearance of which has been attributed to the Jahn-Te l le r effect [2]. Some years ago Coulson and Strauss estimated the vibronic potential energy surfaces of CH4 +, CF4 +, NH3 + and NH3 by the use of the He l lmann-Feynman theorem and some simple assumptions about the electronic charge distribution [3]. I t was shown that in CH4 + and in an excited state of NH3 +, where an electron had been removed from a bonding orbital, ionization was accompanied by a significant Jahn-Te l le r distortion. In CF4 § and an excited state of NH3, where the orbital degeneracy does not involve bonding electrons, the static distortion was predicted to be much less. However, the static distortion of ~ 0- 6 ev predicted from this model for CH4 + is considerably less than might be expected f rom the width of the photoelectron transition. In this paper we present the results of ab initio calculations of the vibronic potential surface for CH4 +, and its application to the spectroscopy of CH4 +.

2. THEORY

I t iS convenient to discuss the potential surface for an orbitally degenerate electronic state exhibiting Jahn-Te l le r instability in terms of an expansion in powers of nuclear displacements. T h e electronic hamiltonian for an arbitrary

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114 R .N . Dixon

fixed nuclear configuration may be developed as a Taylor expansion in nuclear displacements from a reference configuration of high symmetry. Integration over the electronic coordinates then leads to a vibronic hamiltonian which is a function of the nuclear coordinates only. Unlike the case of a non-degenerate state, this vibronic hamiltonian contains not only totally symmetric terms, but also terms which belong to non-totally symmetric representations of the reference point group.

The ground state of tetrahedral CH4 + is a 2:/12 state, the orbital degeneracy of which can be removed by vibronic perturbations of species e and t2 [4]. Let us choose a basis of real 2T2 component functions which transform as translations along cubic axes. The electronic energies of the three components of the 2T2 state for distorted nuclear configurations can then be expressed as the eigenvalues of the following effective hamiltonian matrix:

IT2, x) IT2, y) IT2, ~)

<T2, xl <T2, yl <T2, ~l

1 , , t Ho + ~(He,a - ~3He,b') Ht2,z Htz,v 1 t ! H ' Ho+~(Hea + ~ / 3 H e b ) H ' t2,z , , t2,x

Ht2,y' Ht2,x' H o - He,a'

(1)

where Ho includes all terms of irreducible representation al, I-Ie,a' and I-Ie,o' are the components of the perturbations of representation e, and Ht2,x', Ht2,y' and Ht2,z' are the components of perturbations of representation t2. The vibrational displacement coordinates for methane transform as al + e + 2t2, and will be taken as the usual normalized symmetry coordinates, with typica! components:

si(~) = �89 (~1 + ~ + ~3 + ~ ) , ]

I r

s3(tz,x) = { (3rl + 3r2 - 3r3- ~r4), (2)

r s4(t2,x) = ~ (~oq2 -- 50:34).

H0 will contain the same harmonic and anharmonic force constants as for a 1.41 state. The non-totally symmetric perturbation terms, complete to second order in displacement coordinates, may be written as:

H'e = 12s2 + 112sls2 + 122(s2s21 e) + 13a(e)(sasa l e) ]

+ 134(O(s3s41e) + 144(*)(s4s4 ]e), (3)

H ' t 2 = 13s3 + 14S4 + ll3SlS3 + 114SlS4 + 123(S2S3 [ t2) + 124(S2S4 [ /2)

+ la3(t)(sasalt2) + laa(t)(s3s4lt2) + 144(t)(s4s4[t2).

In these equations the direct products of degenerate displacement coordinates have been expressed in irreducible form, and the vibronic potential constants l~ t are

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On the ~ahn-Teller distortion of CH4 + 115

defined in terrr, s of products normalized as follows:

(szsz l e,a) = �89 2 - sz,b2),

1 (sisj ] e,a) = - ~ (2Si,zS:,z- si,xS:,x- si,vSj,u),

s2s2[e,b) = S2,aS2,b,

(s~sj I e,b) = : ~ (s , , ,sj ,x- $t,ygj,y),

(s2sl l t2,x) = �89 X/3S2,b)Si,x,

(s2s, lt2.v) = �89 + V~2,b)s,,~,, (s2si I tz,.) = s2,.s,,~,

1 (s3s41 t2,~) = ~ ($3,y$4,Z + S3,z$4,y).

i , j=3 , 4,

i , j = 3 , 4,

i=3, 4,

i=3, 4,

i=3, 4,

(4)

There may therefore be three first-order Jahn-Teller splitting parameters and 12 second-order parameters. The absolute minimum in the electronic energy of the lowest component of the 27"2 state corresponds to a tetragonal distortion to the point group D~a if He' is the dominant perturbation, or to a trigonal distortion to the point group C3v if Ht2' is the dominant perturbation [5]. (Since the matrices of He' and Ht2' cannot be simultaneously diagonalized, these two energies of distortion are not additive.)

The direct calculation of the vibronic perturbation parameters using Van Vleck perturbation theory requires an extensive sum over virtual states to allow for orbital following. This work has therefore been based on separate calculations of the total electronic energy as a function of s~, s2, s3 and s4 for configurations of symmetry Ta, D2a, C3v and C3v respectively. In each case the energies were fitted to polynomial functions of displacement coordinates.

2.1. Method of calculation A number of ab-initio calculations of energy surfaces for small molecules have

shown that vibrational force constants can be calculated with 10-20 per cent accuracy from LCAO-SCF-MO wavefunctions using extended basis sets of Slater-type orbitals [6, 7]. In this work each Stater-type orbital has been approximated by a linear combination of gaussian functions [8].

In calculating a potential function of many variables using a contracted gaussian basis it is essential to choose the shortest expansion consistent with the desired accuracy. Hehre, Stewart and Pople have suggested that three gaussian functions per Slater orbital are adequate for many chemical problems, the considerable difference between the energy calculated in this manner and the corresponding Slater orbital energy arising mainly from the poor atomic inner shell description which is largely unaltered in the molecule [9] (Method I). Alternatively, 'mixed basis' calculations have been proposed, in which one-electron integrals, some of which are very sensitive to the detailed form of the orbitals, are exactly calculated over a Slater basis, but the much larger number of more difficult two-electron integrals, which are less sensitive to the detailed form of the orbitals, are approxi- mated using a contracted gaussian basis [10] (Method II). Finally, it would

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116 R . N . Dixon

appear possible to remove much of the error in the pure gaussian calculation with a small basis by calculating all one-centre one and two-electron integrals directly over Slater orbitals, but using a contracted gaussian basis for all one and two- electron many centre integrals (Method III). This will restrict the errors arising from an incomplete gaussian representation of Slater orbitals to the small energy contributions which arise from molecule formation, while avoiding errors in the much larger one-centre energy contributions, which may vary significantly with interatomic distances because of the variation in the degree of overlap of orbitals on different centres.

These various approximations to accurate calculations over a Slater orbital basis have been compared for ' minimal basis ' calculations on tetrahedral CH4 as a function of bond length, using standard orbital exponents for carbon, and ~= 1-2 for hydrogen. For convenience the calculations with five gaussian functions per Slater orbital and Method I I I are taken as a reasonable approximation to exact integration over Slater orbitals. Th e results of these preliminary calcula- tions (table I) show that for small gaussian expansions: (i) Methods II and I I I

Method'j"

II

III

Gaussian basis:~

2, 3/2, 3 3, 3/3, 3 5, 5/5, 5

5, 5/2, 3 5, 5/3, 3

2, 3/2, 3 3, 3/3, 3 5, 5/5, 5

Etotalw A.U.

- 39"6407 - 39"7100 -40-0754

-40.1217 -40-1008

-40.0990 -40.1095 -40.1133

e(lal)w A.U.

- 11"1386 -11"1096 -- 11.2549

-- 11"2875 -- 11"2670

- 11-2290 - 11-2681 - 11"2709

E ( 2 a l ) w

A.U.

-0 .9190 -0"9283 -0"9313

-0"9300 -0-9272

-0.9122 -0.9330 -0.9320

E ( l t 2 ) w

A.U.

-0.5213 -0-5380 -0.5413

-0.5478 -0.5400

-0.5181 -0.5405 -0.5416

f l l rmin. mdyn/

i i

1-112 6.93 1-112 6.77 1.108 6-75

1"229 1.42 1.126 6-09

1-110 6.73 1-111 6.70 1.108 6.74

t See text. Number of gaussian functions per Slater orbital Is, 2s and 2p (one electron-integrals)/Is,

2s and 2p (two-electron integrals). w rc~ = 1 '094 .~ = r0(exp).

Table 1. A comparison of calculated properties of CH4 for various methods of calculation. (Minimal basis.)

do indeed substantially correct the deficiencies of the all gaussian calculation of Method I, such that the calculated energies--and also the electron density matrix elements--are closer to the Slater orbital values (the total energy for an exact Slater calculation with slightly different orbital exponents is - 4 0 . 1 1 2 9 A.U. [11]); (ii) the ' mixed basis ' calculation of ,Method II gives a very poor representation of the variation of the total energy with bond length whereas the force constant is almost independent of the number of gaussian functions for Method III . All three methods gave poor results with only two gaussian functions for each Slater orbital, because of the poor representation of the (ls, 2s) overlap density on carbon and the (ls, ls) hydrogen-hydrogen overlap densities.

The double zeta calculations reported in this paper were therefore carried out using Method III , with three gaussian functions for each ls Slater function, and

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On the Jahn-Teller distortion of CHa + 117

two gaussian functions for each 2s and 2p function (42 gaussian functions contracted to 18 Slater orbitals). The orbital basis consisted of Clementi's recommended double zeta basis for carbon ls, 2s and 2p orbitals [12], and the functions (ls, ~ = 1.39) and (2s, ~ = 2-10) for each hydrogen atom taken from the work of Arrighini et al. on the ground state of CH4 [13]. A check on the reliability of the calculations was provided by comparing the calculated structure and force constants of neutral CH4 with experiment, and by extending the gaussian basis for the equilibrium nuclear configurations of CH4 and CH4 + to three gaussian functions for each Slater orbital (54 gaussian functions in all). Each open-shell calculation for CH4 + made use of Roothaan's formulation [14] of the symmetry restricted hamiltonian for the appropriate point group and component state, although convergence was improved in some cases by using McWeeney's transformation of this hamihonian [15].

3. RESULTS

3.1. Tetrahedral symmetry

The minimum energy for the 1A 1 ground state of CH4 was found to be - 4 0 - 1 6 5 5 A.U. at a bond length of 1-079 ~,. Extending the gaussian basis at this bond length lowered the energy by 0-0134A.U. (table 2). The experimental

Point group

(distortion)

Td

Tel

D2a (s2)

D4h (s2)

Td(Cav hamiltonian)

C3v

C3v (S4)

Structure

r = 1 "079 s x 4)

r = 1 " 1 4 7 s

r=1"147 A ( x 4 ) / _ H C H = 141-2~ x 2) / _ H C H = 9 6 "3~ x 4)

r = 1" 147 A( x 4)

r=1"147 A(x4) / H C H = 109-5~ x 6)

r = 1 " 3 4 5 A(x 1) r=1"081 A(x 3)

/ H C H = 109" 5 ~ x (6)

r = 1 " 1 4 7 Ax (4) / H I C H i = 9 6 "9~ x 3)

/ H C H = 118 "6~ x 3)

CH4 1A 1

- 4 0 . 1 6 5 5 ( - 4 0 " 1 7 8 9 ) t

- 4 0 - 1 5 2 9

- 4 0 . 0 8 6 7 ( - -40 .1028)#

CH4 +

2T z

2 B 2 or 2A1 2 E

- 39.6472 ( - 3 9 " 6 5 7 9 ) t

- 39.6584

- 39.7101 - 39"5512 ( - 3 9 . 7 2 3 5 ) t ( - 3 9 - 5 6 5 2 ) t

- 3 9 . 8 8 7 9

- 4 0 - 1 3 1 1

- 4 0 " 1 3 2 0

- 39.6765 - 39.3539

- 39.6659 - 39"6588

- 39.6829 - 39"6226

- 39.6864 - 39.6221

t 54 gaussian funct ions contracted to 18 orbitals. All other calculations, 42 gaussian funct ions contracted to 18 orbitats.

Table 2. A summary of calculated energies for CHa and CHa + for s tructures at which one of the energies is stat ionary (a.u.)

equilibrium bond length is 1-085 ~ [16], and the calculated symmetric stretching force constant is about 10 per cent higher than the experimental value (table 3). The minimum energy for the 2T2 state of CHa + was found to be -39-6584 A.U.

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at r = 1 " 1 4 7 ~ , compared with -- 39 . 6472 A.U. at r = 1 . 0 7 9 ~ and a value of - 39. 6216 A.U. at r = 1,079 ~ obtained from the ground state wavefunetion using Koopman's theorem. In both the 1A1 and 2T2 states there is a significant anharmonicity, leading to calculated values of f 1 1 1 = - 2 . 2 and - 1 . 3 mdyn/A z respectively. The principal object of this work is to explore the Jahn-Tel ler splitting, and all calculations for non-tetrahedral CH4 and CHa + were carried out with the above determined mean ionic state bond length of 1. 147 ~,.

C o n s t a n t

jell

f~2

f44 lz la 14 122 laa(t) /44 (t)

CH4, 1A1 Calc. Obs.t

6.45 5 "84 0"5055 0"486 4-565 5-38 0.5132 0-458

CH4 +, 2T2 Calc.

5 "05 0"429 4"12 0"422 0"432

-0"577 O" 261 O" 004

-0"145 - 0"022

Unit

mdyn/3. mdyn/~. rndyn/3, mdyn/.~ mdyn mdyn mdyn mdyn/,~ mdyn/h mdyn/A

Table 3.

t Reference [17]. Calculated at r = 1"147 A.

Calculated and observed force constants.

3.2. Distortions to point group D2a

The theory given is w 2 shows that the minimum energy for distortion by s2 will occur for one of the three equivalent D2a point groups, for example by excitation of s2, a. Only eight of the 24 operations of Ta are retained in D2a, and the orbital degeneracy of a 2T2 state is split, leading to 2B2 and 2E states. Separate calculations were carried out for these two states using symmetry restricted hamiltonians for the configurations (1 al) 2(2al) 2(1 e)4( 1 b2) 1 and (1 al ) 2(2al ) 2(1 e)3(1 b2) 2 respectively. T h e energies for tetrahedral CH4 + were identical using either the Ta or D2a hamiltonians as long as care was taken to ensure that for the D2g hamiltonian the orbital containing the unpaired electron was directed along one of the cubic axes. Hence no difficulty was experienced in the continuity of the calculated energies in passing from the point group Ta to D2a. (However, see below.)

The minimum energy of the 2B2 state ( - 39. 7101 A.U.) was found to occur at a strongly distorted nuclear configuration, with two H C H angles of 141.2 ~ and four of 96.3 ~ (figure 1), corresponding to a stabilization energy of 0. 0517 n.u. (1" 41 ev). Extension of the gaussian basis lowered the total energy by 0.0134 A.U., the change being identical to that for extension of the basis for the 1A1 state of neutral CH4 at its equilibrium configuration.

3.3. Distortions to point group C3v

The minimum energy for distortion by s3 and s4 will occur for one of the four equivalent C3v point groups, for example by distortion parallel to the (1,1,1) cubic

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On the Jahn- Teller distortion of C H a + 119

-39"50

-39 '55

E ( a . u . )

-39-60

-39 '65

-39 .70

I I I i

i / i1

zB2 /' q /

~ i I

~ I~ I

J

. // j l

X

z E

E ~ j

\ .

2B 2

i p

I I I I

so ~ ioo ~ moo . i o ~ l e o ~ mo~

Angle HI CH~,

Figure | . Variation of the total electronic energy of CH4 + with an s2(e-type) distortion to the point group D2a (and D ~ ) . The axis of the distortion bisects /_ HICH2(s2a).

open-shell S.C.F. calculations; virtual energy from CH4 calculation using Koopman's theorem.

axis. Six of the operations of Td are retained in C3v, and the orbital degeneracy of 2T2 is split to ~A1 + 2E. Separate calculations were carried out for these two states using C3v symmetry restricted hamiltonians for the configurations (la1)2(2a1)Z(le)a(3a1) 1 and (la1)2(2a1)2(le)3(3a1) 2 respectively. In contrast to the results with a D2a hamiltonian, the energy of the Zdl component for tetrahedral CH4 + was lower by 0.0075 A.U. than that with the Ta symmetry restricted hamiltonian, whereas the energy of the partially restricted 2E component was only 0.0004 n.y. lower than with the T~ hamiltonian. This inconsistency is inherent in the use of the symmetry-restricted S.C.F. open-shell hamiltonian. This has the symmetry of the one-electron density rather than that of the nuclear conformation, although this only leads to difficulties when the open shell M.O. belongs to the same representation as fully occupied M.O.'s, as in 2A1 of C3v. A further mani- festation of this inconsistency is the presence of very small but non-zero linear terms in the dependence of the mean energy of the configuration on s3 and sa, but neither of these terms contribute more than 0-002 n.u. over the range of interest. Since the 2T2 state lies about 0 .4 n.y. lower than the first excited state of CHa + ((lal)2(2a1)t(lt2)6; 2A1), and both sa and s4 lead to Jahn-Teller splittings of ~ 0.06 A.U. at the distortion to the minimum in the 2Az energy, it is assumed

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that the above inconsistencies will not contribute seriously to errors in the calculated Jahn-Tel ler parameters. A check on this assumption was provided by noting that the calculated cross sections through the potential surface w6re approximately parallel to those calculated using Koopman's theorem. This problem could have been avoided by using a single hamiltonian restricted for the mean of the configuration for both the Ta and Csv point groups. However, this would necessarily have given a calculated energy of the distorted 2A1 state higher than that for the best single determinant wavefunction, leading to uncertainties in the determination of the minimum in the energy surface.

- : 5 9 - 5 5

- 3 9 " 6 0

E(o.u)

- ~ 9 " 6 5

i = i i

2

z / / 2 E

~ x r

% /

2 F " ' .

- 3 9 - 7 0 I I =

- 0 . 1 5 0 " 0 0 - 1 5 0 . : 5 0

st, (~) Figure 2. Variation of the total electronic energy of CH4 + with an s3(t2-type) distortion to

the point group C3v. The axis of the distortion is parallel to the bond CH1 (Srl = - 33r2, 3, 4). - - open-shell S.C.F. calculations ; . . . . . . virtual energy using Koopman's theorem.

The maximum stabilization energy with respect to distortion by s3 was 0.0170A.U. with r 1 = 1 . 3 4 5 ~ , r2=rs=r4=l'081.~ (figure 2), and by sa was 0.0205 A.U. with / H1CH2, 3, 4= 96" 9 ~ (figure 3). These two stabilization energies are additive to second order in displacement coordinates if the small off diagonal force constant f12 is ignored, so that the stabilization by both s3 and s4 amounts to 0.0375 A.U. (1.02 ev). Thus even if the complete increment of 0. 0074 A.U. arising from the removal of the Ta symmetry restriction is added to this energy, thereby referring the energy of the 2A1 state to the symmetry restricted 2/'2 energy, the maximum stabilization for a C3v conformation (0.0449 A.U. = 1.22 ev) is lower than that for a D2a conformation (0-0517 A.U. = 1-41 ev). The equilib- rium nuclear configuration of the ground state of CH4 + is therefore predicted to belong to the D2a point group.

For all three non-totally symmetric distortions the contributions to the Jahn-Tel ler splitting from the quadratic parameters li~ were found to be an order of magnitude less at the energy minimum than that from the linear parameters li.

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- 3 9 . 5 5

- 3 9 " 6 0

E (o.u.)

- 3 9 - 6 5

o~

2 E " , . p

I I I I 39-7O 9 0 ~ I 0 0 ~ 110 ~ 1 2 0 ~

A n g l e H n C H 2

Figure 3. Variation of the total electronic energy of CH4 + with an s4(t2-type) d i s to r t i on to the point group Csv. The axis of the distortion is parallel to the bond CH1. - - open-shell S.C.F. calculations; . . . . . virtual energy using Koopman's theorem.

It should be noted that only three of the 12 second-order Jahn-Teller parameters (equation (3)) have been calculated because of the limited number of sets of S.C.F. calculations.

4. DISCUSSION

The calculated quadratic force constants for neutral CH4 (table 3) are all within 15 per cent of the observed values. (The poorest agreement--15 per cent error-- is for faa, for which the use of r = I . 147 3, instead of I . 094 A will have introduced the largest contribution from anharmonicity.) I t is therefore anticipated that the probable errors in the force constants for the 27"2 state of CH4 + are less than _+ 20 per cent. More reliable results would undoubtedly necessitate (i) longer gaussian expansions or direct integration over Slater orbitals, (ii) the addition of polarization functions such as 3d orbitals on carbon and 2/) orbitals on hydrogen, as in Arrighini et al.'s work on CH4 [12] and Karl and Csizmadia's work on CH3 + and CH3- [17], and (iii) the removal of the restriction to single configurational wavefunctions. Extension (iii) would appear to be the best method of avoiding problems in the continuity of the potential function with change in point group.

The prediction that the minimum energy of CH4 + occurs for a D2~ nuclear configuration is in contrast to that of Coulson and Strauss [3] whose semi-empirical calculation led to significant stabilization to Car by sa and s4, but a very small stabilization by s2. I t is interesting in this respect to compare these calculations with those of a simple Hiickel model for the angular distortions s2 and s4. Let the C(2p)-H(ls) resonance integral be taken as proportional to the overlap integral, neglect H - H interactions and mixing between C(2s) and C(2p), and put ~(C, 2/))= ~(H, ls). This simple model leads to a minimum energy of the 2B2 state of Dza

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CH4 + at / H I C H 2 = 141.1 ~ a minimum energy of the 2A 1 state of C3v CH4 + at /_ HICH2 = 100.0 ~ and predicts that distortion by s4 will only lower the energy by 30 per cent of that for distortion by s2. These simple predictions closely parallel the S.C.F. calculations, but not those of Coulson and Strauss.

The potential curve of figure 1 indicates that the energy of square planar CH4 + is about 0.018 A.U. (0.49 ev) lower than that for tetrahedral CH4 +. It therefore seems probable that asymmetrically substituted methanes would freely racemize upon ionization.

4.1. The photoelectron spectrum of methane

The adiabatic and vertical ionization potentials for methane calculated as the difference in S.C.F. energies of the molecule and the ion are 12- 39 ev and 14-18 ev respectively, compared with experimental values of 12.70 ev and about 14-4 ev respectively [2]. Although these calculations are only poor approximations to Hartree-Fock calculations, the error in the calculated ionization potential is comparable to that calculated by Cade and Huo for CH with a very extensive basis set ( - 0 - 5 6 ev, [6]), and may be attributed to the smaller magnitude of the correla- tion energy for the 9-electron CH4 + system compared with the 10-electron CH4 system.

In the D2a point group of the minimum in the potential surface one component of ve (yea) is totally symmetric. The principal vibrational structure near the 0-0 transition of the photoelectron spectrum should therefore consist of a long progres- sion in this vibration, the frequency of which is calculated from the potential curve of figure 1 to be ~ 1300 cm -1. Price [2] has observed differences of ~ 1200 cm -~ between successive vibrational peaks near the origin which are therefore assigned to v2 (cf. v2 = 1526 cm -1 for CH4). At higher transition energies the photoelectron spectrum shows little regular structure, presumably because of the high density of vibrational levels resulting from the excitation of all the normal modes.

A rigorous calculation using the potential function of table 3 of the expected vibrational intensity distribution in the photoelectron spectrum requires a knowledge of the vibronic energy levels and wavefunctions. Unfortunately this presents a formidable problem, involving three strongly coupled Schr6dinger equations in eight vibrational displacement coordinates. However, a meaningful comparison between prediction and experiment may be based on the use of simplified models.

(i) Excitation of Vl The difference in calculated bond length for CH4 + and CH4 (0-072 A) is about

1.5 times the ground state zero-point amplitude of 0.053 h. vl should therefore be excited as a short progression of about four members.

(ii) Excitation of v2

If the orbital degeneracy of a Tz state is only significantly removed by distortions of e symmetry the potential surface consists of three separate (disjoint) sheets, each of which is approximately harmonic in S2a and s2b about one of the three equivalent minima. All the vibronic levels are triply degenerate (tl or t2 levels) [1], the three components corresponding to vibration in the three separate sub-states. A T2+-A1 transition would then be accompanied by a simple progression in one component of v2, the progression having a single intensity maximum. In the limit

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of strong coupling to v2 the progression will exhibit an approximately gaussian intensity distribution with a total width at half intensity equal to 2(ln 2)1/2/2(se 2)t/2, where the average is over the vibrational wavefunction of the v"= 0 level of the At state. The calculated value of this half-width for CH4 + is

2(In 2)1t212<S22>1/2= 2(ln 2)l/212(hcoJ2"/f22")l/2= 1.15 ev. (5)

(iii) Excitation of v3 and v4 Consider first the strong coupling of a 7"2 state to a single t2 mode. The

potential surface for the static distortion consists of a single sheet of three inter- secting branches, the splitting between which depends on both the magnitude and phase of the distortion. Figure 4 is based on the diagonalization of the hamiltonian matrix of equation (1) for a given I ql but variable phase angles 0 and ~b, retaining only the linear perturbation terms, and shows that over much of the normal coordinate space the three branches are split by approximately + l] q I, O, - l l q I about H0 as mean. The three vibronic Schr6dinger equations cannot be uncoupled

I-5

Figure 4. distortion of length q (0 ~< ~b ~< 90 ~ only).

. . . . . ~=30 ~ or 60~ . . . . . . . . ~=45 ~

i i l i i

1"~ I ~.,-~. ~ . . . . . -A-~-~ ~ - .

tq 0 " 5 I - / l ' . . . .

/ . f "..-..

0.0 p - ~ ~ ~..-~ ~"~--

"-2:. .2 \- . . . . . . ;. - 0 . 5 \ . \ / /

- I ' 0 - - '~- - '~" ~ ' ~ q~ . .~--..~-.-

_,..t , . . . . . . . , ,

o ~ ~ o ~ e o ~ 9 0 ~ , z o ~ , 5 0 ~ i s o ~

0 Varia t ion wi th 0 and 4' of the f i rs t -order static spl i t t ing of a T s ta te by a t2-type

- - ~b=0 or 9 0 ~ ~ ~b= 15~ or 75~

by any transformation, and we will make use of the ' semi-classical Franck-Condon approximation' [1, 18]. This depends on the fact that for a band system involving extensive vibrational excitation most transitions end in highly excited vibrational states, for which the vibrational wavefunctions ~b(q) oscillate rapidly except near the classical turning points at which E = V'(q). Hence for a given E only these regions of q-space contribute appreciably to the appropriate transition moment integral, and we may write

~b'(E, q) ~_ 3(E-V'(q)). (6)

Since the vibrations in the excited state are being treated essentially classically, their energy levels are to be regarded as continuous. Thus for excitation of a single

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124 R.N. Dixon

non-degenerate vibration the transition probability P(E) as a function of transition

energy is given by: P(E) ae~ 1 l 2 ~(E-V'(q)) dq, (7)

and the intensity distribution is a convolution of the potential function of the final state over the vibrational probability distribution for the initial state. Equation (7) may be generalised to many dimensions, and for the present problem of a potential function of three branches Vi and a t2 mode we obtain:

3

P(E) aE~ E I ~"(qx, @, @)12 8lE-V/(qx, @, q~)l aqx aqu aq~ i = 1

3

--- Y~ I r 0, r z 8lE_V((q ' O, r z sin 0 dq dO de. (8) i = 1

Figure 5 presents intensity distribution for transition from the v" = 0 level calculated using this model for two sets of parameters. In both cases f ' =f", and the Jahn- Teller parameter is given in the dimensionless form

L= l/(hcojf)l/2, (9)

L = 2

I I I I~ ~ .

i i t "h t

Transition

Probability

L=4

- 6 - 4 - 2 o 2 a s s 0o

AE/hcca Figure 5. Transition probability distribution for a T+-A transition in which the T state is

vibronically coupled to one tz mode, calculated using the semi-classical Franck- Condon approximation. L is the dimensionless first-order Jahn-Teller coupling parameter. The contributions from the three components of the T state are indicated separately.

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On the Jahn-Teller distortion of CH4 + 125

corresponding to the reduced displacement coordinate

9=q(f/hcco)l/2. (10)

In these units the most probable value of Q is Qm = 1, and (Q2)={ . These distributions show three equally spaced maxima, despite the considerable angular dependence of V( upon # and 6, successive maxima being separated by approximately

AE= LQrahcoJ = lqm = I(~(q2> )1/% (11)

For very large L the semi-classical distribution would become symmetrical [19] but for small L the quadratic term in Vi' causes the high energy peak to be poorly resolved from the middle peak.

Sturge [1] has compared the semi-classical approximation with the results of a rigorous numerical treatment of the dynamic Jahn-Teller effect [20] for the simpler case of an E state interacting with an e mode. In this case both models lead to the prediction of a vibronic profile with two maxima, one above and one below the vertical excitation energy for q= 0. For an e mode the most probable value of Q is Qm=2-1/2, and (Q2>=l . The peak-peak separation using the semi-classical model is 2LQmhcoJ=2lqm=2I(�89 and there is a marked minimum between the two maxima; whereas the rigorous calculation shows a lower contrast between maxima and minima, and a slightly larger splitting which approximates to 2l(q2>1/2 over the range 5~<L2~< 30. These differences may be attributed to the neglect of the dynamic coupling of the motion over the two branches of V, which will be greatest for q ~ 0 or E ~ Eo. The semi-classical model will similarly lead to an underestimate of the splitting of a T2 state, and equation (10) is therefore replaced by the empirically increased predicted peak-peak separation:

AE = l(q2> 1/2. (12)

Two t2 modes couple to the 2 T2 state of CH4 +. For such a case it may be shown that a linear transformation of the normal coordinates can lead to concentration of the Jahn-Teller coupling into one displacement coordinate, and that the predicted peak-peak separation is then found to be:

AE= [la2(s32) + 2131a(s3s4) +/42(s42)] 1/2

= 0.705 ev (13)

(iv) Excitation of all normal modes

The coupling of the T2 state to v2 is not strictly independent of that to v3 and v4, but will be approximately so since /2 does not contribute to the same matrix elements (equations (1) and (2)) as/3 and/4. Thus the potential of table 3 leads to a predicted intensity distribution having three broad equally spaced maxima separated by about 0.7 ev with the highest energy peak being poorly resolved, the structure arising from coupling to v~ and v4 and the breadth from coupling to vl and v2. The photoelectron spectrum has just this appearance [2], with broad peaks of low contrast at about 13.6, 14-4 and 15-0 ev. This close agreement between experiment and calculation confirms the Jahn-Teller interpretation of the structure of the spectrum, and shows that the calculated potential function cannot be seriously in error.

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126 R . N . Dixon

4.2. A 2E<-2B2 absorption system of CH4 +

Herzberg [21] has suggested that the diffuse absorption lines observed in the spectra of distant stars, of which the broad feature at 4430 h (2.8 ev) is the best known example, [22], are produced by the predissociated spectrum of a polyatomic molecule present in interstellar gas, and that this molecule may be CH4 +. Under interstellar conditions the molecules would be predominantly in the lowest level of the ground state, and such an absorption would arise from the allowed ~E+-2B2 transition in the point group D2d. The calculated vertical transition energy (figure 1) is 4-3 ev. The 2E state will be subject to Jahn-Teller interaction with the bl component of v2 (v2b) and the b2 components of v3 and v4 (vaz and Vaz), leading to an intensity distribution over these vibrations which is estimated to have two maxima separated by 1.2 ev. In addition, excitation of a long progression in V2a(al) will further broaden the intensity distribution, giving a calculated width at half intensity of 1-8 ev. The 2E- 2B2 transition of CHa + would therefore be expected to lead to absorption throughout the visible and near ultra-violet, with a complex vibrational structure arising from strong interactions between the normal modes, and could therefore well account for the observations.

I should like to thank Dr. D. B. Cook for help with the use of routines from the Polyatom molecular integral programme, and Professor R. McWeeney (both of Sheffield University) for the use of his IBM 1130 computer, on which all the calculations were carried out.

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Spectrosc. Ion Phys., 1, 285. PRICE, W. C. (private communication). [3] COULSON, C. A., and STRAUSS, H. L., 1962, Proc. R. Soc. A, 269, 443. [4] HERZBERO, G., 1966, Electronic Spectra and Electronic Structure of Polyatomic Molecules

.. (D. Van Nostrand Co. Inc.), p. 50. [5] OPIK, U., and PRu M. H. L., 1957, Proc. R. Soc. A, 238, 425. MOFFITT, W., and

THORSON, W., 1957, Phys. Rev., 108, 1251. [6] CADF., P. E., and Huo, W. M., 1967, ft. chem. Phys., 47, 614. [7] KaRI, R. E., and CSIZM~a)IA, I. G., 1969, ~. chem. Phys., 50, 1443. [8] BoYs, S. F., 1950, Proc. R. Soc. A, 200. HUZINACA, S., 1965,y. chem. Phys., 42, 1293.

O-OHATA, K., TtUCSTA, H., and HUZINAGA, S., 1966, ~. phys. Soc. ~apan, 21, 2306. [9] HEHRE, W. J., STP.W~mT, R. F., and POPLP., J. A., 1968, Symposium Faraday Soc., 2, 15.

[10] COOK, D. B., and PALMmm, P., 1969, Molec. Phys., 17, 271. [11] ARRmHINI, G. P., M~mSTRO, M., and Moccm, R., 1967, Chem. Phys. Lett., 1, 242. [12] CLEMENT1, E., 1964, J. chem. Phys., 40, 1944. [13] ARRICHINI, G. P., GUn)OTTI, C., MAESTRO, M., MoccIa, R., and SALV~.TTI, O., 1968,

3. chem. Phys., 49, 2224. [14] ROOTH~N, C. C. J., 1960, Rev. mod. Phys., 32, 179. [15] McWEENY, R., 1964, The Self-Consistent Generalization of Hiickel Theory, Molecular

Orbitals in Chemistry, Physics and Biology (Academic Press Inc.). [16] KUCHITSU, K., and B~mTELL, L. S., 1962, ft. chem. Phys., 36, 2470. [17] DUNC~_~, J. L., and MILLS, I. M., 1964, Spectrochim. Acta, 20, 523. [18] HImZBERO, G., 1950, Spectra of Diatomic Molecules (D. Van Nostrand Co. Inc.), p. 392 [19] TOYOZAWA, Y., and INotm,..M., 1966, ft. phys. Soc. ~apan, 21, 1663. [20] LONCtmT-HICCINS, H. C., OPIK, U., PRYC~, M. H. L., and SACK, R. A., 1958, Proc. R.

Soc. A, 244, 1. [21] HF~'~ZB~C;, G., 1967, I.A.U. Symposium, 31, 91. [22] HEm3IG, G. H., 1967, I.A.U. Symposium. 31, 85.

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