The nature of molecular vibrations selected by various excitation processes

Chemical Physics 138 (1989) 237-244 North-Holland, Amsterdam

THE NATURE OF MOLECULAR VIBRATIONS SELECTED BY VARIOUS EXCITATION PROCESSES Henrik Grum KJAERGAARD and Ole Sonnich MORTENSEN Fysisk Institut, Odense Universitet, DK-5230 Odense hf. Denmark Received 17 April 1989; in final form 7 August 1989

Local modes of vibration have been clearly identified in infrared absorption, but sofar not in other kinds of optical spectroscopy. We have investigated the reason for this through model calculations on a simple, but realistic, double Morse oscillator model of the stretch vibrations of a CHs radical, excited by various optical processes involving a change of electronic state. The general result is, that the resulting spectrum is determined mainly by the excitation operator, and is normal-mode-like for excitation processes dominated by Franck-Condon factors. The specitic form of the spectrum depends on the details of the excitation operator, and we investigate how the characteristics of the normal-mode-like spectrum depend on the difference in the equilibrium distance of the two potential surfaces.

1. Introduction

The local mode picture is now well established in describing the overtone spectra of various molecules containing CH moieties [ l-41. Even with large simplifications in the calculations, like excluding the bending motion and approximating the true potentials by a sum of Morse potentials, the local mode picture has given convincing results [ 5-7 1. Recent calculations by, e.g., Halonen and Carrington [ 81 have improved the numerical agreement between theory and experiment, in calculations where both the bending motion and the coordinate dependence of the Wilson G-matrix [9] are included. It has been possible to interpret the spectra and assign the peak positions, whereas there are still difficulties with the peak intensities. The reason for this is that the dependence of the electric dipole moment on the nuclear coordinates is still not well known. Local modes of vibrations have so’ far only been identified in near infrared (NIR) spectra but not in other forms of spectroscopy, e.g., electronic absorption or Raman dispersion (Radis) spectra. There is no reason why local modes should not be present in electronic excited states. The reason that they are not seen in the spectra must then be due to the excitation

operator, which presumably selects primarily states that are non-local in nature. This question has already been considered by Heller and Gelbart [ 10 1. They made model calculations of the electronic emission spectrum from a system consisting of two non-coupled Morse oscillators having a somewhat larger equilibrium distance in the excited than in the electronic ground state. Even though this system represents a pure local mode limit, they showed that the emission spectrum had the form of a typical normal mode spectrum due to the symmetric stretch normal mode. Heller and Gelbart [ lo] did not investigate, how the characteristics of this pseudonormal mode spectrum, e.g., the apparent frequency, anharmonicity and change in equilibrium distance, depended on the parameters of the Morse oscillator, nor did they consider other optical processes than emission. In this paper we attack essentially the same question as addressed by Heller and Gelbart, but through a more realistic model situation, and look at both absorption and Raman dispersion spectra. We investigate in detail how the parameters of the apparent normal mode spectrum depend on the parameters of the effective excitation operator. The plan of the paper is as follows. In section 2 we give a short review of the curvilinear normal mode

0301-0104/89/$03.50 0 Elsevier Science Publishers B.V. ( North-Holland Physics Publishing Division )

238

H.G. Kjaergaard, OS. Mortensen /The nature ofselected molecular vibrations

and the local mode picture of molecular vibration. This section is included to establish notation and set the stage for the following calculations. Section 3 gives expressions for cross sections used to calculate the various spectra. In section 4 the resulting spectra are shown and analysed with particular emphasis on the relation of the parameters of the apparent spectral Hamiltonian to the parameters of the true double oscillator system. A short conclusion is given in section 5.

2. Normal and local mode pictures of molecular vibration

(1)

V,=V+

;x+ ?I

potential

H=$

1 G$pipj+ ij

I’, defined as:

,GI_l/'Goy), J

(2) where in eq. (2) the differential operator, a/aqt, only operates on what is inside the brackets, and (G ( is the determinant of the Wilson G-matrix [ 9 1. Vis the potential energy for the vibrational motion. The volume element for integration is simply nidq,. The definition of the matrix elements G, [ 121 shows that the kinetic energy part of the Hamiltonian is a rather complicated function of the internal coordinates. As pointed out by Sibert et al. [ 131, it is a fairly good approximation to calculate the G, for a

P(qi 2..*, qn) >

(3)

with pj= - iria/aqF This approximate expression is exact for stretching vibrations with bending neglected, since the G-matrix elements are then constant. Expanding the potential energy in powers of q allow us to write the vibrational Hamiltonian as: H=f

Molecular vibration is best treated using curvilinear internal displacement coordinates, qt. These are changes from equilibrium values of bond lengths and valence angles. The internal displacement coordinates are favourable because the potential energy to a good approximation can be written as a sum of terms depending on only one internal displacement coordinate. The first problem is then to transform the nuclear kinetic energy from Cartesian coordinates to internal coordinates. This problem was first solved by Podolsky [ 111, and recently we [ 12 ] have rederived it in a straightforward way using only simple mathematics. After transformation the Hamiltonian is:

with the (slightly modified)

fixed nuclear configuration, e.g., equilibrium. The second term in eq. (2) is then zero, and the Hamiltonian reduces to

1 Gz,p:+$ I

C G$pipj i#i

+ 4 1 Fiiqf + 4 1 FoqiqJ I i+j + 1

,fia +

f,nmndia ,

(4)

I

where the Fare the well-known force constants. The last two terms in eq. (4) are the higher-order diagonal and off-diagonal terms in the potential energy, respectively. Since the Schrijdinger equation with the Hamiltonian of eq. (4) cannot be solved exactly, we divide the Hamiltonian into a zeroth-order Hamiltonian, Ho, which defines the basis, and a perturbation H’. The difference between the curvilinear normal mode picture, and the local mode picture, lies then with the separation of the Hamiltonian. In the curvilinear normal mode picture the first and second rows of eq. (4) defines Ho. The rest is then in principle H’, but in practice one rarely goes beyond fourth order in q. We use the term curvilinear normal mode picture to emphasize that the coordinates indeed are curvilinear, but the form of the normal coordinates expressed as linear combinations of internal coordinates are the same for rectilinear as for curvilinear coordinates. In the local mode picture we take the first column in eq. (4) as Ho and the zeroth-order Hamiltonian can be written as a sum of one-dimensional Hamiltonians: Ho= 1 hi(P‘, qi) 3 I with h, = 4 G$pf + Ifi The local mode

potential

. I’i(qi)

(5) will typically

be

H.G. Kjaergaard,O.S. Mortensen/The nature ofselected molecularvibrations

Morse-like for stretching vibrations and harmonic for a bending vibration. As the perturbation we take the harmonic coupling terms: H’=f

iCjGtPiPj+f

(6)

,zjF,qiqj,

that is all higher-order non-diagonal terms in V are neglected, as they empirically are known to be small. In the local mode picture we thus have harmonically coupled anharmonic oscillators. The basis states are eigenfunctions of Ho and simple product functions of the form: Iv*Q...~,)=

101)

I~2)..*IU,)

.

(7)

If the molecule contains more than one oscillator of a particular type the eigenvalues will be degenerate and it is convenient to use symmetrized eigenstates of Ho as basis states. The Hamiltonian matrix then becomes block diagonal, one block for each symmetry. From now on we specialize to a realistic but simple model of a symmetric molecule with bending neglected. We use Morse potentials for the two equivalent stretching modes. The zeroth-order Hamiltonian, which we solve exactly, is given by: HO=~G~,p~+D[1-exp(aq,)]2

+tG?1p:+D[1-exp(-uq2)]2,

(8)

with the perturbation (9)

H1=G%P,P2+F,2q,q2.

D is the dissociation energy and a the usual Morse

scaling parameter. We introduce the dimensionless parameters [ 5 1, 1

G12

1

F,2 (10)

y=-zJ@yq2’

“=2JF,F,’

to characterize the kinetic and potential energy coupling respectively. The Hamiltonian for two harmonically coupled identical Morse oscillators can then be written as: (H-E”,>)/hcv’=v-(v:+v:+v)x +rta:

-6)

(0:

-a21

+@(a:

+al)

(az’

+a2)

239

tional excitation, and states which have the same v are said to belong to the same manifold. Ey,,,,‘,,is the zeroth-order energy of the ground state. The wavenumber, a, and the anharmonicity, x, are the common parameters for the two equivalent stretching modes. a + and a are the creation and annihilation operators respectively. Even though they here operate on Morse oscillator functions these operators have, to a good approximation, the usual step-up and step-down harmonic properties [ 5 1. We call this the ladder approximation. Coupling between different manifolds may, to a good approximation, be neglected, since the manifolds are well separated in energy and the parameters y and @are small for the systems of interest here. The Hamiltonian matrix will then be block diagonal with one block for each manifold. As shown previously [ 5 ] the calculations hereby become very simple, and the resulting eigenenergies are in good agreement with experimental NIR spectra.

3. Excitation processes and transition operators As mentioned in section 1 we will consider two different excitation processes namely one-photon absorption and two-photon resonance Raman scattering. Both involve two different electronic states. In the dipole approximation the absorption cross section for the transition g-+f is given by [ 14 ] :

(12) where (Yis the usual fine-structure constant, r, is the level width of the final state, R, is the dipole matrix element between states g and f, fir*= ( e8 - Ef) / hc and e is the polarization of the photon field. For the two-photon Raman scattering we introduce the differential scattering cross section (do/dL?), defined as [ 141: (13)

,

(11)

where v= vi + y is the quantum number of vibra-

Here cS is the wavenumber of the scattered photon, d&2is the solid angle and the scattering tensor, +,, is defined as:

240

H.G. Kjaergaard, OS. Mortensen /The nature of selected molecular vibrations

(14) where the summation goes over all possible intermediate states. v’, is the wavenumber of the incident photon and &,, p0 are the components of the dipole operator. In the following we use adiabatic wavefunctions. The initial and final molecular states are written: g=Vi(r,

4),&v(4) 7 f=Vj(r7

CTIXjw(4) >

(15)

where indices i and j refer to electronic quantum numbers and index V,w to vibrational quantum numbers. The electronic and nuclear wavefunction is denoted by t,~ and x, respectively. r, q stand for electronic and nuclear coordinates, respectively. In the adiabatic approximation the dipole matrix element becomes:

where from now on ( ) means integration over all nuclear coordinates and ( ) means integration over all electronic coordinates. We expand pi/(q) in a Taylor series about the equilibrium nuclear configuration. Retaining only the constant term (Condon approximation) the expression for the relative absorbance becomes:

where (Xj~lX~~) is the overlap integral between the two vibrational states ;ci,,,and xiy, so the square is the usual Franck-Condon factor. The Condon approximation is appropriate for electronic transitions where i#j. To describe purely vibrational transitions in the NIR region one must of course go beyond that. If one restricts the expansion to the linear term, the transition operator will then be linear in the local coordinates, and will connect only states that differ by one or more quanta in a single local coordinate. This is precisely the reason that local mode states are preferentially excited in NIR absorption. In a resonance Raman dispersion (Radis) experiment one measures the cross section as a function of

the incident photon wavenumber. Near resonance, 0, x Fig, the second term in the scattering tensor, ( 14)) is small compared to the first, hence it can neglected. The relative cross section divided by then becomes, in the Condon approximation:

&za

s

c e

IXiwlXje> (XjeIXiv> i&-q-ire

’ *

i.e. eq. be ij:

(18)

The intermediate states, e = wjxje belong to the electronic excited state, whereas the ground and final state belong to the electronic ground state. To calculate a Radis spectrum we use eq. ( 18 ) with fixed final state, Xiw,and varying B1.

4. Results and analysis In NIR spectra the experimental data [ 51 show a tendency to favour “pure” local mode state, I vO), that is states where all the excitation is localized in a single bond. In the following we will show that the picture is somewhat different for Franck-Condon excitations. The calculations are done on a simple but realistic molecular system: the CH2 fragment, neglecting the bending vibration. This has earlier been shown to be a resonable approximation in interpreting the NIR spectra of the dihalomethanes [ 51. The calculations are done within the local mode picture. The wavefunctions are according to eq. (7) the product of two Morse oscillator wavefunctions [ 15 1. Couplings between manifolds are neglected and the so-called ladder approximation is used, thus the Hamiltonian is given by eq. ( 11). The parameters we use are appropriate to a C-H bond in the electronic ground state of CH&: ~=0.923 amu, ij=3143.4 cm-‘, x=0.0201 and y-+0.0099. The same parameters are used in the electronic ground and excited state, except for a displacement, Ar, in the equilibrium distance of the two C-H bonds. In all spectra shown in the following the wavenumbers are measured relative to the vertical wavenumber difference between the two vibrational ground states. A common level width, r, is used for all vibrational levels. This level width is varied to show the effects of varying resolution and media effects on linewidths. In fig. 1 a Franck-Condon excited absorption spectrum is shown at high resolution, r=50 cm-‘.

H.G. Kjaergaard, OS. Mortensen / The nature of selected molecular vibrations

0

5000

10000 T7

15000

(cm-’ )

Fig. 1. Absorption spectrum for CHs fragment with Ar=0.3 A and at high resolution, r= 50 cm-‘. Representative states are labeled with their local mode state designation.

8

s

n

s

s

0

20000

10000

241

tits the broadened two oscillator spectrum is also shown as the dashed curve in fig. 2. The single Morse oscillator spectrum is characterised by four parameters, the oscillator wavenumber and anharmonicity, the displacement and the level width. These four parameters are found from a leastsquares fit of the “observed” double oscillator spectrum. As fig. 2 shows the agreement between the two spectre are quite convincing. The wavenumber and anharmonicity of the single Morse oscillator can be simply found from a standard Birge-Sponer plot, that is a plot of [E(v) -8( 0) l/v against v. Such a plot is shown in fig. 3 and it is seen that the deviations from the expected linear behaviour are within the “experimental” uncertainty. Neither the wavenumber nor the anharmonicity of the single Morse oscillator are the same as those of the true local oscillators, but are functions of the displacement, Ar. This dependence is illustrated in figs. 4 and 5. The point of the curves corresponding to Ar= 0 shows the wavenumber and anharmonicity of the true local oscillator. It is seen that both curves have two limits corresponding to very small and rather large displacements, respectively. At small displacements the parameters of the single oscillator tend towards those of the local oscillators, whereas for larger displacements both anharmonicity and wave-

30000

3000 1

< (cm-l) Fig. 2. Absorption spectrum for CHs fragment with Ar=0.3 A and r= 500 cm- *, solid curve. Single Morse oscillator absorptionspectrumwitb~=3027cm-‘,x=0.00964,~=775cm-’and Ak0.4358 A, dashed curve.

Representative states are labeled with their symmetrized local mode state designations, I uIv2) f = 2-‘j2( )v,v2) + 1v,v2) ). Transitions to anti-symmetric states are of course forbidden by symmetry. The spectrum shows all the allowed transitions and also that the various bands within a single manifold have quite different intensities. At low resolution typical of condensed systems, r= 500 cm-‘, we obtain the spectrum shown in fig. 2 (solid curve). The spectrum now resembles that of a single oscillator, rather than that of a double oscillator. The single Morse oscillator spectrum that best

‘; E U > . z 0

2900 -

2800 -

IW I >

2700 -

rw’

2600

’ 0

’ 2

’

’ ’ ’ & 6

’

’ 8

’

’ ’ 1 10 12

V

Fig.3.Plotof[I!?(v)-8(0)]/uagainstvuse.dtofindtheMome parameters of the apparent single oscillator spectrum showed as solid curve in fg. 2.

H.G. Kjaergaard, O.S. Mortensen /The nature ofserected molecular vibrations

242

r-_

o.020

L

sorption spectra the important term is ( v] 0” ) , where the double prime indicates the electronic ground state. To define the transition operator we write:

0

(u(O”)=(v~T~O)+T~0)=(0”).

0 0

The Hamiltonians cited states are:

0 OO

0

0

0

H”=K”+V”,

0.005 1

-1

0.1

0.2

0.4

0.3

ground and ex-

(20)

where K is used for the kinetic energy. Because K” = K we have: withA=V”-V.

If the displacement

Ar (A) Fig. 4. Variation of the anharmonicity of the apparent single mode oscillator seen in calculated absorption spectra of CH2 fragment for different values of displacement.

for the electronic

H=K+V,

H”=H+A, 0.0

(19)

loft)=

(21)

is small so is A, hence:

Io)+i;o~

Ii).

(22)

By comparison with the standard solution of the perturbation equations [ 161 we find:

3150

Txl+ 3100

0

0 0

0

0

0 0

3000 E 2950 0.0

0.1

0.2

Ar

0.3

0.4

(A)

Fig. 5. Variation of the wavenumber of the apparent single mode oscillator seen in calculated absorption spectra of CH2 fragment for different values of displacement.

number of the single oscillator is significantly lower than those of the local oscillator and in fact are seen to converge towards well defined values. The results for small displacements can be understood by a comparison with the NIR spectra. In these the important term is (~)p( 0)) with the transition operator T=/r, which to a good approximation is linear in q1 and q2. In the Franck-Condon excited ab-

l-P, -A, E,-H

(23)

where we use 1= &Pi and that Pi is the projection operator for the state ( i) . For small displacements, A is approximately linear in q1 and q2, thus T is linear in the coordinates. Hence the states selected are the pure local mode states, in complete analogy to the situation in NIR spectra. In figs. 4 and 5 we see that Y”and x for the apparent single oscillator approaches those of the original local mode oscillator. To understand the large displacement limit we transform the Hamiltonian, eqs. (8) and (9), from local coordinates to the symmetry coordinates (curvilinear normal modes ) : a,=-jp+yl),

u-=-$(or42)~

The Morse potential nite sum:

in eq. (8 ) is written as an in&

(24)

V(q,)=D[l--xp(-aq,)l’ =D f

(-l)‘%&zq’)’

(25)

I=2

and similarly Hamiltonian:

for

V(q2).

This

leads

to

the

243

H.G. Kjaergaard, O.S. Mortensen /The nature of selected molecular vibrations H(~+~-)=~(G~,+G~~)P:+~(GPI-G~~)PZ_ + tw,,

+F,,)q:

-(@)

+ 1 (F** --F12)q2_

oa3(q:

+&Du”(q:

+3q+qC)

+6q:q2-

This Hamiltonian terms:

.

+qf)+...

can be written

(26)

as a sum of three

+H’(q+,q-).

(27)

The coupling Hamiltonian H’ is large in contrast to the situation in the local mode picture. The Hamiltonian for the q+ coordinate describes to a very good approximation a Morse oscillator with the potential:

(28)

~(q+)~=!~(1-exp[-((alJZ)q+l)*. The wavenumber

is slightly changed:

V+

=P(l-y+@)x3110cm-‘, whereas the anharmonicity factor of two:

(29) essentially

number is significantly lower than expected. Also the displacement is found to be very close to that of the q+ oscillator as seen in table 1. The fact that the wavenumber is not well described by the q+ oscillator is an indication of the strong mixing of the basis states. In a resonance Raman dispersion spectrum (Radis), the picture is somewhat different. The spectrum is sensitive both to the displacement and to the final state chosen. Fig. 6 shows a calculated Radis spectrum for the I 10) + final state. The spectrum show single mode behaviour although it is not as clearcut as in the absorption spectra. The parameters for the single oscillator cannot be found from a BirgeSponer plot of a Radis spectrum. This is due to the fact that the cross section for Radis essentially is a square of a sum and not a sum of squares and the resulting interference affects both peak positions and peak heights. The single oscillator parameters can however be found from a least-squares fit of a calcuTable 1 Displacement and linewidth of the single Morse oscillator spectra that best fit the calculated spectra for different values of the displacement, Ar. All values are found by a least-squares fit

decreases by a

hcv”, x+=4x2D _ = ; +

$

x0.495xx0.00995

.

Similarly the displacement increases with the factor ,/2 in the q+ coordinate, whereas the displacement for the q_ coordinate is identically zero. As said the coupling term is large. Therefore if we choose a basis of products of “normal” anharmonic oscillator functions: 1v+ v_ ) = 1v+ ) I v_ > , the eigenfunctions of the Hamiltonian will involve heavy mixing of the basis functions. If we for the moment neglect this mixing, then Franck-Condon excitation from the ground state, 1O+O_ ), will lead to states of the form I v+O_ ) since the displacement in the q_ coordinate is zero. Therefore in the absence of mixing the spectrum would be that of a single Morse oscillator with the parameters given above. As figs. 4 and 5 show the anharmonicity is in good agreement with that of the q+ oscillator but the wave-

Ar (A)

Ar, (A)

Arf Ar,

r, (cm-‘)

0.05 0.1 0.2 0.3

0.0702 0.1420 0.2852 0.4358

1.40 1.42 1.43 1.45

500 500 595 775

*VI

I,

C

u

‘0

u

0

10000

20000

30000

3 (cm-l) Fig. 6. Radis spectrum for CH, fragment with 1f) = 110)+, r=550 cm-‘, and Ar=0.3 A.

244

H.G. Kjaergaard, OS. Mortensen /The nature of selected molecular vibrations

lated single-oscillator spectrum to the two-oscillator spectrum. When this is done it is found that the parameters are reasonably close to those found from the absorption spectrum. In general Radis spectra are more complicated than absorption spectra, however, and so it is far more difficult to draw simple conclusions.

in the resonance Raman dispersion spectra they may also show more complicated behaviour. To show the “true” eigenstate of the molecule high-resolution spectra are necessary. References [ 1] B.R. Henry, in: Vibrational Spectra and Structure, Vol. 10, ed. J.R. Durig (Elsevier, Amsterdam, 198 1) pp. 269-3 19.

5. Conclusion

[ 21 M.S. Child and L. Halonen, Advan. Chem. Phys. 57 ( 1984) 1.

[ 31 MS. Child, Accounts Chem. Res. 18 (1985) 45.

The results described in this paper indicate how careful one should be in interpreting vibronic spectra at low resolution. As shown, a double mode Hamiltonian often leads to low resolution spectra which resemble spectra of a single oscillator. Thus they might be described by a simple one-dimensional spectral Hamiltonian, rather than the double mode vibrational Hamiltonian. The spectral Hamiltonian, however, often has a complicated connection with the full vibrational Hamiltonian. The observed band maxima do not necessarily coincide with the energies of the molecular eigenstates. Hence the states selected by the transition operator are generally not eigenstates of the true Hamiltonian but rather wavepackets. Depending on the transition operator these wavepackets may have local character as seen in the NIR spectra and the small-displacement Franck-Condon excited absorption spectra, or normal character as shown in the large-displacement Franck-Condon excited absorption spectra. As seen

[4] B.R. Henry, Accounts Chem. Res. 20 (1987) 429. [ 51 O.S. Mortensen, B.R. Henry and M.A. Mohammadi, J. Chem. Phys. 75 ( 1981) 4800. [6] M.S. Child and R.T. Lawton, Faraday Discussions Chem. Sot. 71 (1981) 273. [7] B.R. Henry, A.W. Tarr, O.S. Mortensen, W.F. Murphy and D.A.C. Compton, J. Chem. Phys. 79 (1983) 2583. [ 81 L. Halonen and T. Carrington Jr., J. Chem. Phys. 88 ( 1988) 4171. [9] E.B. Wilson, J.C. Decious and P.C. Cross, Molecular Vibrations (McGraw-Hill, New York, 1955 ). [lo] E.J. Heller and W.M. Gelbart, J. Chem. Phys. 73 ( 1980) 626. [ 111 B. Podolsky, Phys. Rev. 32 (1928) 812. [ 12 ] H.G. Kjaergaard and O.S. Mortensen, Am. J. Phys., in press. [ 131 E.L. Sibert, J.T. Hynes and W.P. Reinhardt, J. Phys. Chem. 87 (1983) 2032. [ 141 O.S. Mortensen and S. Hassing, in: Advances in Infrared and Raman Spectroscopy, Vol. 6, eds. R.J.H. Clark and R.E. Hester (Heyden, London, 1980) pp. l-60. [ 151 L. Pauling and E.B. Wilson, Introduction to Quantum Mechanics (McGraw-Hill, New York, 1935). [ 161 E. Merzbacher, Quantum Mechanics (Wiley, New York, 1970).