Development of the Hamiltonian, matrix elements, and computer

Development of the Hamiltonian and matrix elements for partially deuterated methoxy radical

DMITRY MELNIK†, JINJUN LIU‡, ROBERT F. CURL*† AND TERRY A. MILLER*‡

†

Rice Quantum Institute and Chemistry Department, Rice University, Houston, TX 77005, USA

‡

Laser Spectroscopy Facility, Department of Chemistry,

The Ohio State University, Columbus, Ohio 43210-1173, USA

Robert Curl: [email protected] Terry Miller: [email protected] Dmitry Melnik: [email protected] Jinjun Liu: [email protected]

1

ABSTRACT The effects of partial deuteration of the methoxy radical upon the ground state Hamiltonian are considered. Methoxy exhibits Jahn-Teller distortion and has an unpaired electron somewhat complicating the situation. Two approaches are considered. One named the internal axis method sets up the rotational and spin-rotation Hamiltonians with the axis of quantization aligned along the direction of the orbital angular momentum. The other named the principal axis method chooses the axis of quantization along the aaxis. The internal axis method is preferred because the perturbation Hamiltonian needed to treat CH3O is simply expressed and has simple matrix elements with this choice.

2

1. Introduction What happens when the symmetry of a Jahn-Teller molecule is broken by isotopic substitution poses intriguing possibilities. From the point of view of the chemical dynamicist, CHD2O may provide a gateway into interesting photochemical dynamics such as rearrangement into hydroxymethyl[1] or dissociation into formaldehyde and H[2], as the barrier to either process is not dauntingly high. This may be accomplished either through stepwise excitation of the CH stretch of CHD2O or its direct excitation to an overtone of the CH stretch. To the spectroscopist on the other hand understanding the molecular Hamiltonian, the nature of the rovibronic energy levels, and the vibrational and rotational spectra provide exciting new intellectual challenges. Since very little progress has been made so far in obtaining and unraveling the vibrational spectrum of the normal isotopomer, the place to start gaining this understanding is the rotational structure of the vibrational ground state. Jet-cooled moderate resolution laser induced fluorescence (LIF) measurements were first done by Kalinovsky[3]. We have recently improved the LIF accuracy and have obtained extensive spectra of CHD2O and CH2DO with standard deviations of measurement of about 50 MHz[4]. Stimulated emission pumping (SEP) spectra probing the upper fine structure component are being obtained with similar accuracy. Jet-cooled high resolution spectra of the vibrational ground state have been obtained through microwave spectroscopy (lines of both CHD2O and CH2DO have been measured with sub-MHz accuracy)[5]. Thus experimental energy levels of the ground vibrational state of these unsymmetrically deuterated isotopomers are near to hand. An approximate theoretical model to analyze the moderate quality LIF data was proposed by

3

Kalinovsky;[3] however, the newly available high resolution data require more detailed developments of the theory. This work is aimed at filling this need. First symmetry labeling will be discussed. Then two different approaches to setting up the Hamiltonian will be described and compared. These approaches are reminiscent of the two methods developed long ago for the treatment of single symmetric rotor internal rotation and consequently could be given the same names, internal axis method and principal axis method, except that they might somehow be confused with the internal rotation problem, which they actually resemble only slightly. Therefore, they are called the internal axis treatment and principal axis treatment. The two axis systems used for the two treatments are respectively denoted (x,y,z) and (a,b,c) and are illustrated in Fig. 1.

2. Symmetry For the undeuterated, CH3O, and totally deuterated, CD3O, isotopomers the molecular symmetry group is isomorphous with C3v. This results in the rovibronic levels being labeled with A1, A2, and E symmetry labels and the vibronic levels being labeled with the double group symmetry irreducible representations, A1, A2, E, E1/2, and E3/2 except that only E1/2 and E3/2 representations appear preserving Kramers degeneracy. The partially deuterated isotopomers of interest here have a molecular symmetry group isomorphous with Cs. The rovibronic levels can be labeled A' and A". In going from C3v symmetry to Cs symmetry, the correlation is A1→A', A2→A", and E→A'+A". Again the vibronic levels have to be labeled by the double group symmetry, in this case the single irreducible representation E1/2, which preserves Kramers degeneracy. For the two isotopomers, the molecular symmetry group has the two elements, E and (12)*. When 4

(12)* is broken into a C2 operation affecting the orientation of the molecular frame with respect to space fixed axes and a σ plane affecting only coordinates in the molecule-fixed frame, the orientation of the elements with respect to the principal axes differs between the two isotopomers. For CHD2O, the elements comprising (12)* are C2b and σac; for CH2DO, the elements comprising (12)* are C2c and σab.

3. Case (a) basis functions The ground state of CH3O has traditionally been treated in case (a) as its energy level pattern is close to case (a). There is no reason to change from that basis. The basis functions used for CH3O have been | Σ, P, J , p〉 =

1 | Λ = +1, Σ, P, J 〉 + (−1) J - P + S - Σ + p | Λ = −1, −Σ, − P, J 〉 ) ( 2

(1)

Here p,which can take on the values of 0 or 1, is called the parity though that is a misnomer. In this basis, there is a separate Hamiltonian matrix of size 2(2J+1) for each value of J and p. This does not make maximum use of the symmetry of CH3O. For example for J=1/2, this results in a 4×4 Hamiltonian matrix for each value of p. The resulting energy levels consist of three distinct E symmetry levels and an A1 symmetry level for p=0 and the matching three E symmetry levels and one A2 level for p=1. The same basis functions can be used for CHD2O and CH2DO provided that the yaxis is chosen perpendicular to the plane of symmetry (y matches b for CHD2O and y matches c for CH2DO). For these unsymmetrical isotopomers, there is no further symmetry factoring with p=0 corresponding to A' levels and p=1 to A" levels.

5

4. CH3O Hamiltonian and matrix elements The effective Hamiltonian and matrix elements of the ground state of CH3O were developed in several seminal papers [6-9]. These results are conveniently summarized in the book by Hirota [10]. This effective Hamiltonian of the ground vibronic state was derived beginning with the rotational angular vector operator in the form R=J-L-G-S where R is the angular momentum of nuclear rotation, J is the total angular momentum, L is the electronic orbital angular momentum, G is the vibrational angular momentum, and S is the electron spin. Then a Van Vleck perturbation treatment is applied [8]. This results in an effective Hamiltonian that no longer contains the components of L and G, Lx, Ly, Gx, Gy, that are off-diagonal in vibrational and electronic states, adds some new terms [8], and modifies most of the Hamiltonian parameters. In order to add the effects of asymmetric deuterium substitution, we will need to revisit parts of the treatments leading to the effective Hamiltonian. In the present section, our purpose is to describe the effective Hamiltonian of CH3O and to describe how its matrix elements are calculated. For the ground state of CH3O, the effective Hamiltonian is of the form H = HSO + HCOR+ HROT + HSR + HROT,PERT+HSR,PERT

(2)

where HSO is the spin-orbit interaction, HCOR is the interaction of z-component of rotation with the orbital angular momentum, HSR is the spin-rotation interaction, HROT,PERT and HSR,PERT result from the combination of the Jahn-Teller interaction and the Van Vleck perturbation interaction with excited electronic states as described by Hougen.[8] The effective Hamiltonian is in terms of the components of N=J-S and the pseudo operator, L,

6

which is defined by the properties Lz|Λ〉=Λ|Λ〉 and L ±2 | Λ = ∓1〉 =| Λ = ±1〉 . The terms in the effective Hamiltonian are HSO=aζedSzLz

(3)

HCOR=-2AζtNzLz

(4)

HROT=ANz2 + B(N+N-+N-N+)/2

(5)

HSR=εzzNzSz + (εxx+εyy)(N+S-+N-S+)/4

( with εxx=εyy)

H ROT , PERT = h1 ( L -2 N +2 + L +2 N −2 ) + h2 ⎡⎣L -2 ( N z N − + N − N z ) + L +2 ( N z N + + N + N z ) ⎤⎦

(6) (7)

H SR , PERT = ε1 (L -2 N + S + + L +2 N − S − ) + ε 2 a ⎡⎣L -2 ( N z S − + S − N z ) + L +2 ( N z S + + S + N z ) ⎤⎦ +ε 2b ⎡⎣L -2 ( N − S z + S z N − ) + L +2 ( N + S z + S z N + ) ⎤⎦

(8)

To elaborate on Eqs. (3) and (4), prior to the perturbation treatment, the spin-orbit interaction is HSO = aLzSz; afterwards it is written HSO=aζedLzSz where ζe is a quenching arising from interaction with excited electronic states, d is orbital quenching resulting from the Jahn-Teller effect [6]. The Coriolis term that was originally -2ANz(Lz+Gz) is modified to -2AζtNzLz. All the other parameters in the Hamiltonian are affected by the perturbation treatment, but are symbolized by their familiar forms. Note that the Coriolis term arises from the substitution Rz=Nz-Gz-Lz in the original rotational Hamiltonian; this will arise later. Eqs. (7) and (8) are introduced by the Van Vleck perturbation [8]. Now substitution of N=J-S gives the Hamiltonian appropriate for a case (a) basis. For HSO this substitution has no effect. For HCOR this substitution gives -2Aζt(Jz-Sz)Lz, which splits into a Coriolis term, -2AζtJzLz, plus a correction to aζed, i.e. modifying HSO. For HROT, this substitution gives, neglecting additive constants that will only shift the vibronic energy, 7

HROT=A(Jz –Sz)2 + B(J+J-+J-J+)/2-B(J+S-+J-S+)

(9)

For HSR, this substitution gives, again neglecting additive constants, HSR=εzzJzSz + (εxx+εyy)(J+S-+J-S+)/4

(10)

Note that the cross-terms between J and S in HROT are of exactly the same form as those of HSR. When the substitution is made in HROT,PERT, one formula that results is the same as the formula above for HROT,PERT with N substituted by J. The terms in J+S+ that arise from the substitution in the h1 part of HROT,PERT have the same structure as the terms of HSR,PERT involving ε1 with N substituted by J. The quadratic terms in S±2 resulting from both h1 and ε1 give zero for S=1/2. The terms involving products of J and S that result from the substitution N=J-S in the h2 terms of HROT,PERT give rise to terms that have either the same structure as those of ε2a or the same structure as those of ε2b. The final structures of the form SiSj that result from both HROT,PERT and HSR,PERT give zero for S=1/2 and case (a). Thus if we somewhat redefine what we call spin-orbit and Coriolis, and what we call rotational and what we call spin-rotational, we can collect all the terms into equations. H'SO=(aζed+2Aζt)SzLz

(11)

H'COR=-2AζtJzLz

(12)

H'ROT=AJz2 + B(J+J-+J-J+)/2

(13)

H'SR=(-2A+εzz )JzSz +[-B+ (εxx+εyy)/4](J+S-+J-S+)

(14)

H 'ROT , PERT = h1 ( L -2 J +2 + L +2 J −2 ) + h2 ⎡⎣L -2 ( J z J − + J − J z ) + L +2 ( J z J + + J + J z ) ⎤⎦

(15)

8

H 'SR , PERT = ( −2h1 + ε1 )(L -2 J + S + + L +2 J − S − ) + (− h2 + ε 2 a ) ⎡⎣L -2 ( J z S − + S − J z ) + L +2 ( J z S + + S + J z ) ⎤⎦ + ( − h2 + ε 2b ) ⎡⎣L -2 ( N − S z + S z N − ) + L +2 ( N + S z + S z N + ) ⎤⎦

(16)

All these expressions are easily evaluated when applied to the basis functions of Eq. (1). We now go through this process of reorganizing the Hamiltonian in order to make clear how the same sort of process will work when applied to the new terms that arise upon partial substitution.

5. Added terms for partially deuterated CH3O 5.1Vibronic quenching The breaking of C3v symmetry by partial deuteration attempts to quench the orbital angular momentum and would succeed if the spin-orbit interaction were zero. The two resulting electronic states, one of symmetry A' and the other of symmetry A", would be of the form | A'〉 =

1 (| Λ = +1〉+ | Λ = −1〉 ) 2

(17a)

| A"〉 =

1 (| Λ = +1〉− | Λ = −1〉 ) 2

(17b)

Effectively A′ and A′′ correspond to diabatic electronic basis functions with the half-filled pπ orbital respectively along the y and x axes in Fig. 1. The Jahn-Teller problem is normally considered in a basis consisting of these diabatic functions times 2D harmonic oscillator eigenfunctions for each of the Jahn-Teller active mode, and an effective H developed for this basis. In this approach, the non-Jahn-Teller active

9

vibrations are ignored. This is reasonable as their inclusion would only shift the energies of all the above basis functions by a constant amount. However with partial deuteration the situation becomes more complicated. In general the frequencies of the non-Jahn-Teller active modes will differ for the A′ and

A′′ surfaces. This can be easily envisioned by an over-simplified example that nonetheless captures most of the principles involved. Suppose that the deuteriums of CHD2O were much, much heavier than the hydrogen, effectively reducing, for example, the methyl umbrella motion to a one dimensional H motion. For the A′ surface the H would be moving towards and away from a half-filled pπ orbital localized on the O. For the A′′ state the corresponding motion would involve the filled pπ orbital . Clearly the two surfaces would result in different vibrational frequencies for the H motion and correspondingly different zero-point energies. In the partially deuterated methoxy it is necessary to account, in the effective Hamiltonian, for the now non-constant energy shifts due to the different zero-point energies of the A′ and A′′ potentials. The simplest way to achieve this effect is to introduce effective Hamiltonian terms of the form HQ = −

δE 2

(| Λ = −1〉〈Λ = +1| + | Λ = +1〉〈Λ = −1 |)

(18a)

which can be rewritten using the Hougen pseudo operators as: HQ = −

δE 2

(L

2 +

+ L −2 )

(18b)

When HQ is applied to basis functions like those of Eq. (1), its non-zero matrix elements will be 10

〈−Σ,−P,J, p | HQ | Σ,P,J, p〉 = −(−1) J −P +S−Σ+ p

δE 2

(19)

5.2Spin-orbit, coriolis, rotational and spin-rotational asymmetry The breaking of C3v symmetry has possible effects upon all of these terms. What new terms should be included in the Hamiltonian depends strongly upon the choice of the molecule-fixed axis system in which the problem is to be set up. Here we describe two choices of molecule-fixed axis system and the relationship between them.

6. Internal axis treatment The orbital angular momentum of CH3O is aligned along the C3 axis as is the rotational a-axis. Partial deuteration alone is not sufficient to quench completely the orbital angular momentum, which is now no longer aligned with the principal axes of the rotational tensor. Since it is straightforward to set up the rigid rotor in any axis system and it is less easy to understand treatment of the orbital angular momentum when no axes are in alignment with it, the simple thing to do is to align one of the rotational axes with the direction of the orbital angular momentum making the quantum numbers Σ and P of the basis functions refer to the same direction as the quantum number Λ. With this choice of axis, H'SO is unchanged, but the Coriolis, rotational, and spin-rotational terms must be modified. It is not obvious and will be discussed below, but the form of neither H'ROT,PERT nor H'SR,PERT is affected by the introduction of asymmetry, but, of course, the value of the constants h1, h2, ε1, ε2a and ε2b resulting from fitting of the spectrum may be different in magnitude. This leaves the effects of the introduction of asymmetry upon the form of the effective Hamiltonian exclusively in HROT, HCor and HSR. 11

6.1Rotational Hamiltonian In the internal axis approach, the z axis is aligned with the direction of the orbital angular momentum and the x axis is perpendicular to it in the symmetry plane, which is the ac plane for CHD2O and the ab plane for CH2DO (see Fig. 1.). Starting with H ROT = Pxx Rx2 + Pxz (Rx Rz + Rz Rx ) + Pzz Rz2 + Pyy Ry2

(20)

all cross-terms resulting from the substitution R = N-G-L lead to off-diagonal matrix elements in vibrational or electronic energy but those involving Gz+Lz. The off-diagonal terms are treated by the Van Vleck perturbation and lead to no terms new in form. However, in the cross-terms involving Gz+Lz, Gz+Lz must be replaced by ζtLz. Thus in the internal axis approach, the nuclear rotation energy is expressed as H ROT = Pxx N x2 + Pxz (N x N z + N z N x ) + Pzz N z2 + Pyy N y2

(21a)

H COR = −2 Pxzζ t N xL z − 2 Pzzζ t N zL z

(21b)

where the orbital angular momentum axis is tilted in the xz plane. Previously we expressed HROT in terms of N+, N-, and Nz. We must recast Eq. (21a) in terms of these quantities. The result is ⎛ P + Pyy ⎞⎛ N + N− + N− N + ⎞ H ROT = Pzz N z2 + ⎜ xx ⎟ ⎟⎜ ⎠ 2 ⎠⎝ 2 ⎝ ⎛ P − Pyy ⎞ 2 ⎛ Pxz ⎞ 2 +⎜ xx ⎟(N + + N− )+ ⎜ ⎟[(N + + N− )N z + N z (N + + N− )] 4 ⎠ ⎝ 2 ⎠ ⎝

(22a)

(22b)

Eq. (22a) is identical in form to HROT of the symmetric molecule Eq. (5). Thus the effect of asymmetry on HROT is contained in Eq. (22b) and must be developed by the replacement Ni=Ji-Si. As before, the terms quadratic in J, give rise to replacement of Ni with Ji giving 12

⎛ P + Pyy ⎞⎛ J + J− + J− J + ⎞ H' ROT = Pzz J z2 + ⎜ xx ⎟ ⎟⎜ ⎠ 2 2 ⎝ ⎠⎝

⎛ P − Pyy ⎞ 2 ⎛ Pxz ⎞ 2 +⎜ xx ⎟(J + + J− )+ ⎜ ⎟[(J + + J− )J z + J z (J + + J− )] 4 ⎠ ⎝ 2 ⎠ ⎝

(23)

The terms bilinear in J and S, give rise to the same form as HSR again. Thus ⎛ P + Pyy ⎞⎛ J + S− + J− S + ⎞ H'ROT ,CORR = −2Pzz J z Sz − 2⎜ xx ⎟+ ⎟⎜ ⎠ 2 ⎠⎝ 2 ⎝

⎛ P − Pyy ⎞ ⎛ Pxz ⎞ −2⎜ xx ⎟(J + S+ + J− S− ) − 2⎜ ⎟[(J + + J− )Sz + J z (S+ + S− )] 4 ⎠ ⎝ 2 ⎠ ⎝

(24)

where use has been made of the fact that all components of J commute with all components of S. The Coriolis Hamiltonian is obtained by substitution H COR = −2ζ t Pxz N xL z − 2ζ t Pzz N zL z = −2ζ t Pxz ( J x − S x )L z − 2ζ t Pzz ( J z − S z )L z

(25)

= −ζ t Pxz ( J + − S + + J − − S − )L z − 2ζ t Pzz ( J z − S z )L z

6.2 Spin-rotation Hamiltonian The effect of asymmetry upon the spin-rotation Hamiltonian is to introduce two new parameters εxx-εyy and εxz. HSR=εzzNzSz + (εxx+εyy)(N+S-+N-S+)/4

(26a)

⎛ε − ε ⎞ ε +⎜ xx yy ⎟(N + S+ + S+ N + + N− S− + S− N− ) + xz (N x Sz + N z Sx + Sx N z + Sz N x ) 2 ⎝ 8 ⎠

(26b)

where again the Eq.(26b) part consists of the terms introduced by asymmetry. The effective spin-rotation is produced by the Ni=Ji-Si substitution. In this case, since there

13

are no contributions from the SiSj aside from additive constants, the result is obtained by replacing Ni with Ji giving HSR=εzzJzSz + (εxx+εyy)(J+S-+J-S+)/4 ⎛ε − ε ⎞ +⎜ xx yy ⎟(J + S + + J− S− ) + εxz (J x Sz + J z S x ) ⎝ 4 ⎠

(27a) (27b)

where again the fact that J commutes with S has been used. When structures of the same form from HROT are included, we have ⎡ ⎛ Pxx + Pyy ⎞ ⎛ ε xx + ε yy ⎞ ⎤ ⎛ J + S− + J − S+ ⎞ H 'SRwROT = (−2Pzz + ε zz )J z Sz + ⎢ −2 ⎜ + ⎥⎜ ⎟⎠ 2 ⎟⎠ ⎜⎝ 2 ⎟⎠ ⎦ ⎝ 2 ⎣ ⎝ ⎛ −2(Pxx − Pyy ) + εxx − εyy ⎞ ⎛ −2Pxz + εxz ⎞ ⎟[(J + + J− )Sz + J z (S+ + S− )] ⎜ ⎟(J + S+ + J− S− ) + ⎜ 4 2 ⎝ ⎠ ⎝ ⎠

(28)

6.3The perturbation Hamiltonian with asymmetry All of the rotational Hamiltonian terms present for the asymmetric molecule were considered by Hougen in his development of the perturbation Hamiltonian for CH3O [8]. These arise because the CH3O molecule is distorted into an unsymmetrical geometry as it visits the three equivalent minima created by the Jahn-Teller effect. This requires the introduction of these unsymmetrical HROT terms in the rotation-vibration treatment of the ground electronic state and into orbit-rotation interaction with the excited electronic state. These two kinds of second order perturbation effects each lead to the same forms, i.e. those found in HROT,PERT. The quantities ε1, ε2a and ε2b arise from the cross-terms between the orbit-rotation and spin-orbit interactions in the second order perturbation treatment of the interactions between the ground electronic state and excited electronic states. Thus there are no new forms generated in the perturbation Hamiltonians in moving

14

from the C3v isotopomers to the asymmetrically deuterated molecules although the magnitudes of h1, h2, ε1, ε2a and ε2b might be different.

6.4Centrifugal distortion Every constant, k, in the effective Hamiltonian can be modified by writing it as k(1- cJJ(J+1) -cKP2) . Often the operators multiplying k change P. Suppose the form of the Hamiltonian term is kO where O is an operator that connects different P states, then the term corrected for centrifugal distortion should take the form k[1-cJJ(J+1)- 0.5cK(JzO+OJz)]. This statement should not be construed as a development of the centrifugal distortion treatment of this class of molecules, but rather as general rule of thumb that usually works. For the kind of very low rotational temperature data presently available, centrifugal distortion effects are so small that it is very difficult to test this proposal. The matrix elements of the Hamiltonian are listed in Table 1. In trying to relate the formulae given in Table 1 to the equations previously derived, note that J+|P,J〉=f(J,P-1)| P-1,J 〉 while S+|S, Σ〉=|S, Σ+1〉. A computer program for least squares fitting ground rotational and spin state energies has been developed and is available.

7. Principal axis treatment We now consider setting up the Hamiltonian in the principal axis system as first suggested by Kalinovsky [3]. As indicated earlier, in the principal axis system, the Cs plane is coincident with the ab-plane for CH2DO and the ac-plane for CHD2O, respectively. Therefore, y-axis is coincident with the c-axis for CH2DO, but with the b15

axis for CHD2O. The a-axis tilts from the C-O bond by a small angle θ (see Fig. 1.). We refer to CHD2O specifically for the remainder of this section. The corresponding result of CH2DO can be obtained merely by interchanging the b and c axes.

Our basis functions will remain in the same form as exhibited by Eq. (1). However there is a subtle difference in that the angular momentum projections denoted in Eq. (1) are now along the a-axis. This will have consequences, both positive and negative, as we develop the form of an effective H.

The vibronic quenching term in the Hamiltonian is treated in exactly the same manner as in the internal axis case (see Eq. 18). The remainder of the total H, as before, can be written as a sum of terms,

H = H SO + H ROT + H COR + H SR + H ROT , PERT + H SR , PERT

(29)

For CH3O, or in the internal axis approach, HSO reduces to an effective H term of the form:

HSO = aζedSzLz

(30)

within the 2E state due to the vanishing within this basis of the matrix elements of Lx, Ly and Lz is the Hougen pseudo operator as before. When Lz is applied to the Eq. (1) basis functions, it gives the result 1, i.e. HSO = aζedSz

(31) 16

Neglecting any change in the electronic distribution relative to the nuclei, it is appropriate to write Sz in the principal axis system, Sz=Sacosθ+Scsinθ

(32)

giving HSO = aζed(Sacosθ+Scsinθ)

(33)

In this form HSO has both diagonal matrix elements within the fine structure states and between them, as Sa and Sc have, respectively, the form of the Pauli matrices Sz and Sx.

Since the asymmetrically deuterated methoxy radicals are asymmetric tops, the rotational H is of course more complicated than it is for CH3O; however it is simpler than in the internal axis system since in the principal axis system the inertial tensor is diagonal. Therefore we can write the rotational kinetic term of H as:

H KE = ARa2 + σ ( Rb2 + Rc2 ) + δ ( Rb2 − Rc2 )

where R=N-(L+G), σ ≡

(34)

B+C B−C , and δ ≡ , and where A, B, and C are the 2 2

rotational constants of the asymmetric rotor. Substitution of R=N-(L + G) and expansion yields,

H KE = A[ N a2 − 2 N a ( La + Ga ) + ( La + Ga ) 2 ] +σ {[ N b2 − 2 N b ( Lb + Gb ) + ( Lb + Gb ) 2 ] + [ N c2 − 2 N c ( Lc + Gc ) + ( Lc + Gc ) 2 ]}

(35)

+δ {[ N b2 − 2 N b ( Lb + Gb ) + ( Lb + Gb ) 2 ] − [ N c2 − 2 N c ( Lc + Gc ) + ( Lc + Gc ) 2 ]}

17

Since the b axis is perpendicular to the Cs plane, due to symmetry, the matrix elements of Lb and Gb diagonal in the ground vibronic state vanish. Furthermore, following the argument of Hougen[8], the purely vibronic operators (Li + Gi)2, where i=a, b, c, contribute only additive constants to a given vibronic states and hence can be omitted. Eq. (35) thus can be reduced to,

H KE = {[ AN a2 + σ ( Nb2 + N c2 )] − 2 AN a ( La + Ga )}

(36)

+{δ ( Nb2 − N c2 ) − 2(σ − δ ) N c ( Lc + Gc )}

which can be broken down into four terms:

sym H ROT = AN a2 + σ ( N b2 + N c2 )

(37a)

sym H COR = −2 AN a ( La + Ga )

(37b)

asym H ROT = δ ( N b2 − N c2 )

(37c)

asym H COR = −2(δ − σ ) N c ( Lc + Gc ) = −2CN c ( Lc + Gc )

(37d)

where the superscripts asym and sym stand for the symmetric and asymmetric terms of the Hamiltonian, respectively. Further substitution of N=J-S gives the explicit expression of the rotational and Coriolis interaction term of the Hamiltonian in the principal axis system:

18

1 sym H ROT = AJ a2 + ( B + C )( J b2 + J c2 ) 2

(38a)

sym H ROT , CORR = −2 AJ a S a − ( B + C )( J b Sb + J c S c )

(38b)

sym H Cor = −2 A( J a − S a )( La + Ga ) = −2 Aζ t cos θ ( J a − S a )L z

(38c)

1 ( B − C )( J b2 − J c2 ) 2

(38d)

asym H ROT , CORR = −( B − C )( J b Sb − J c S c )

(38e)

asym H Cor = −2C ( J c − S c )( Lc + Gc ) = −2Cζ t sin θ ( J c − S c )L z

(38f)

asym H ROT =

In the same way as in the internal axis description, a subscript CORR stands for the rotational terms bilinear in J and S. We also have used the same geometrical relationship between the expectation values of (La/c+Ga/c), ζ ta / c , and ζ t as we applied earlier for ζ ea / c , and ζ e .

The spin-rotation Hamiltonian in the principal axis system should have the same form as it does in the internal axis system. We therefore can write it directly by replacing x, y, and z in equation (26) with c, b, and a, respectively,

sym = ε aa J a S a + ( H SR

ε bb + ε cc 4

)( J + S − + J − S + )

(39a)

ε bb - ε cc

(39b)

asym H SR = ε ac ( J a S c + J c S a ) + (

4

)( J + S + + J − S − )

19

This is the same form for HSR that we have used for the larger alkoxy radicals, e.g., ethoxy and the T conformer of 1-propoxy. Note however that the spin-rotation tensor components are now defined with respect to the principal axis system while previously (Eqs. 26-27) they were defined in the internal axis system. The matrix elements of the Hamiltonian in the principal axis system are listed in Table 2. Correspondingly, a computer program equivalent to that in the internal axis system has also been developed.

8. Relationship between the two treatments For all terms in the effective Hamiltonian except the two perturbation terms, the two treatments are equivalent. This can be shown by considering the relationship between the two sets of rotational constants.

⎛ A + C⎞ ⎛ A − C⎞ 2 2 Pxx = ⎜ ⎟ −⎜ ⎟(cos θ − sin θ ) ⎝ 2 ⎠ ⎝ 2 ⎠

(40a)

⎛ A − C⎞ Pxz = −⎜ ⎟2sin θ cosθ ⎝ 2 ⎠

(40b)

Pyy = B

(40c)

⎛ A + C⎞ ⎛ A − C⎞ 2 2 Pzz = ⎜ ⎟+⎜ ⎟(cos θ − sin θ ) ⎝ 2 ⎠ ⎝ 2 ⎠

(40d)

N x = −N a sin θ + N c cosθ ; N y = N b ; N z = N a cosθ + N c sin θ

(41)

Substituting these expressions into the internal axis expressions for the rotational and Coriolis Hamiltonians, the expressions for the principal axis rotational and Coriolis

20

Hamiltonians are obtained. We have already seen that spin-orbit term is affected in the same way by the tilt. H SO = aζ e dS zL z = aζ e dS z = aζ e d ( S a cosθ + Sc sin θ )

(42)

remembering that we can leave out Lz since it just gives 1 in the Eq. (1) basis. The resulting matrix for HSO alone has the property that the spin-orbit splitting is constant and equal to aζed regardless of the value of θ. Thus omitting out the two perturbation expressions, the two treatments are equivalent. However, the situation becomes complicated when H'ROT,PERT and H'SR,PERT are considered. In the first case in addition to using the expression Ja in terms of θ, J+=Jx+iJy=cosθJc+iJb-sinθJa has to be created and substituted into the expression for H'ROT,PERT. Developing expressions for the matrix elements of the resulting complicated formula for H'ROT,PERT becomes quite messy. A similar situation arises for H'SR,PERT. We have not attempted to carry this out. However, we have set up the matrix for the h1 term in HROT,PERT in the internal axis method for J=3/2 and transformed it analytically by an arbitrary angle β using Mathematica. The resulting 8×8 matrix in terms of trigonometric functions in β is a nasty looking mess. As mentioned before, it seems awkward to have the orbital quantum number, Λ, refer to one axis, z, and the electron spin quantum number, Σ, and the J quantum number, P, refer to a different axis, a. However, the principal axis development works. Finally each approach introduces five new parameters. For the internal axis treatment, these are δE, (Pxx-Pyy)/2, Pxz, (εxx-εyy)/2 and εxz; for the principal axis approach for CHD2O, these are δE, (B-C)/2, θ, (εbb-εcc)/2 and εac, since the spin-rotation tensor and the principal axis tensor need not be aligned absent a symmetry requirement. Because of 21

the complications associated with setting up the HPERT terms, the internal axis treatment seems simpler.

Acknowledgments The authors wish to thank C. Bradley Moore, Vadim L. Stakhursky, Xiaoyong Liu and Vladimir. A. Lozovsky for valuable discussions. The work at Rice University was supported by the Robert A. Welch Foundation. The work at the Ohio State University was supported by the U. S National Science Foundation via Grant No. CHE0511809.

22

References [1] S. Saebo, L. Radom, and H. F. Schaefer, J.Chem. Phys. 78, 845 (1983). [2] N. D. K. Petraco, W. D. Allen, and H. F. Schaefer, J. Chem. Phys. 116, 10229 (2002) [3] I. Kalinovsky, “Laser Induced Fluorescence Spectroscopy of CHD2O and CH2DO and High Resolution Spectroscopy of CH3O and HFCO'', Ph.D. Thesis, U. of California, Berkeley, 2001. [4] D. Melnik, J. Liu, V.L. Stakhursky, X. Liu, V.A. Lozovsky, R.F. Curl, T.A. Miller, C.B. Moore, unpublished data. [5] D. Melnik, V.L. Stakhursky, V.A. Lozovsky, T.A. Miller, C.B. Moore, F.C. De Lucia, 59th International Symposium on Molecular Spectroscopy, WJ09 (2004). [6] M.S. Child, H.C. Longuet-Higgins, Phil. Trans. Roy. Soc. A 254, 259 (1961). [7] J.M. Brown, Mol. Phys. 20, 817 (1971). [8] J.T. Hougen, J. Mol. Spectrosc. 81, 73-92 (1980). [9] X.M. Liu, T.A. Miller, Mol. Phys. 75, 1237-58 (1992). [10] E. Hirota, High-Resolution Spectroscopy of Transient Molecules, Springer-Verlag, Berlin, 1985.

23

Table 1. Hamiltonian Matrix Elements of CHD2O or CH2DO in the Internal Axis System by Parametera Orbital quenching δE/2

〈−Σ,−P,J, p | HQ | Σ,P,J, p〉 = −(−1) J − P + S −Σ+ p

δE 2

Spin orbit aζed 〈Σ, P, J , p | aζ e dS zL z | Σ, P, J , p〉 = aζ e d Σ Coriolis -2Pzzζt 〈Σ, P, J , p | −2 Pzzζ t ( J z − S z )L z | Σ, P, J , p〉 = −2 Pzzζ t ( P − Σ) -2Pxzζt 〈Σ, P + 1, J , p | ζ t Pxz ( J + + J − )L z | Σ, P, J , P〉 = −ζ t Pxz f ( J , P) 〈Σ, P − 1, J , p | ζ t Pxz ( J + + J − )L z | Σ, P, J , P〉 = −ζ t Pxz f ( J , P − 1) Rotation Pzz 〈Σ, P, J , p | Pzz ( J z − S z ) 2 | Σ, P, J , p〉 = Pzz ( P − Σ) 2 ⎛ Pxx + Pyy ⎞ ⎜⎝ 2 ⎟⎠ ⎛ Pxx + Pyy ⎞ (J + J − + J − J + ) ⎛ Pxx + Pyy ⎞ ⎡ J (J + 1) − P 2 ⎤⎦ 〈Σ, P, J, p | ⎜ | Σ, P, J, p〉 = ⎜ ⎟ ⎝ ⎝ 2 ⎠ 2 2 ⎟⎠ ⎣ ⎛ Pxx + Pyy ⎞ ⎛ Pxx + Pyy ⎞ 1 1 J + S− + J − S+ ) | Σ = − , P, J, p〉 = − ⎜ f (J, P) 〈Σ = + , P + 1, J, p | − ⎜ ( ⎟ ⎝ ⎝ 2 2 2 ⎠ 2 ⎟⎠ ⎛ Pxx + Pyy ⎞ ⎛ Pxx + Pyy ⎞ 1 1 J + S− + J − S+ ) | Σ = + , P, J, p〉 = − ⎜ f (J, P − 1) 〈Σ = − , P − 1, J, p | − ⎜ ( ⎟ ⎝ ⎝ 2 2 2 ⎠ 2 ⎟⎠ ⎛ Pxx − Pyy ⎞ ⎜⎝ 2 ⎟⎠ ⎛ Pxx − Pyy ⎞ (J −2 + J +2 ) ⎛ Pxx − Pyy ⎞ f (J, P) f (J, P + 1) 〈Σ, P + 2, J, p | ⎜ | Σ, P, J, p〉 = ⎜ ⎟ ⎝ ⎝ 2 2 2 ⎠ 2 ⎟⎠ ⎛ Pxx − Pyy ⎞ (J −2 + J +2 ) ⎛ Pxx − Pyy ⎞ f (J, P − 1) f (J, P − 2) 〈Σ, P − 2, J, p | ⎜ | Σ, P, J, p〉 = ⎜ ⎟ ⎝ ⎝ 2 2 2 ⎠ 2 ⎟⎠

24

Table 1 (cont’d) Hamiltonian Matrix Elements of CHD2O or CH2DO in the Internal Axis System by Parametera ⎛ Pxx − Pyy ⎞ ⎛ Pxx − Pyy ⎞ 1 1 (J − S− + J + S+ ) | Σ = + , P, J, p〉 = − ⎜ f (J, P) 〈Σ = − , P + 1, J, p | − ⎜ ⎟ ⎝ ⎝ 2 ⎠ 2 ⎟⎠ 2 2 ⎛ Pxx − Pyy ⎞ ⎛ Pxx − Pyy ⎞ 1 1 (J − S− + J + S+ ) | Σ = − , P, J, p〉 = − ⎜ f (J, P − 1) 〈Σ = + , P − 1, J, p | − ⎜ ⎟ ⎝ ⎝ 2 2 2 ⎠ 2 ⎟⎠

Pxz 〈Σ, P + 1, J, p |

Pxz (2P + 1 − 2Σ) f (J, P) J − J z + J z J − − 2Sz J − )| Σ, P, J, p〉 = Pxz ( 2 2

Pxz (2P − 1 − 2Σ) f (J, P − 1) J + J z + J z J + − 2Sz J + )| Σ, P, J, p〉 = Pxz ( 2 2 1 1 〈Σ = + , P, J, p | −Pxz J z S+ | Σ = − , P, J, p〉 = −Pxz P 2 2 〈Σ, P − 1, J, p |

Spin-rotation εzz 〈Σ, P, J, p | ε zz (J z - Sz )Sz | Σ, P, J, p〉 = ε zz (P − Σ)Σ ⎛ ε xx + ε yy ⎞ ⎜⎝ 2 ⎟⎠ ⎛ ε xx + ε yy ⎞ J − S+ ⎛ ε xx + ε yy ⎞ f (J, P) 1 1 〈Σ = + , P + 1, J, p | ⎜ | Σ = − , P, J, p〉 = ⎜ ⎟ ⎝ ⎝ 2 2 2 ⎠ 2 2 ⎟⎠ 2 ⎛ ε xx + ε yy ⎞ J + S− ⎛ ε xx + ε yy ⎞ f (J, P − 1) 1 1 〈Σ = − , P − 1, J, p | ⎜ | Σ = + , P, J, p〉 = ⎜ ⎟ ⎝ ⎝ 2 2 2 2 ⎠ 2 2 ⎟⎠ ⎛ ε xx − ε yy ⎞ ⎜⎝ 2 ⎟⎠ ⎛ ε xx − ε yy ⎞ J − S− ⎛ ε xx − ε yy ⎞ f (J, P) 1 1 〈Σ = − , P + 1, J, p | ⎜ | Σ = + , P, J, p〉 = ⎜ ⎟ ⎝ ⎝ 2 2 2 ⎠ 2 2 ⎟⎠ 2 ⎛ ε xx − ε yy ⎞ J + S+ ⎛ ε xx − ε yy ⎞ f (J, P − 1) 1 1 〈Σ = + , P − 1, J, p | ⎜ | Σ = − , P, J, p〉 = ⎜ ⎟ ⎝ ⎝ 2 2 2 2 ⎠ 2 2 ⎟⎠

εxz

〈Σ, P + 1, J, p | 〈Σ, P − 1, J, p |

ε xz 2

ε xz 2

J − Sz | Σ, P, J, p〉 = J + Sz | Σ, P, J, p〉 =

ε xz 2

ε xz 2

Σf (J, P) Σf (J, P − 1)

ε ε 1 1 〈Σ = + , P, J, p | xz J z S+ | Σ = − , P, J, p〉 = xz P 2 2 2 2 25

Table 1 (cont’d) Hamiltonian Matrix Elements of CHD2O or CH2DO in the Internal Axis System by Parametera HROT,PERTb h1 1 1 〈Σ = + , −( P − 2), J , p | h1L -2 J +2 | Σ = − , P, J , p〉 = h1 (−1) J − P + S −Σ + p f ( J , P − 1) f ( J , P − 2) 2 2 1 1 〈Σ = − , −( P − 1), J , p | −2h1L -2 J + S + | Σ = − , P, J , p〉 = −2h1 (−1) J − P + S −Σ + p f ( J , P − 1) 2 2 h2 1 1 〈Σ = + , −( P + 1), J , p | h2L -2 [ ( J z − S z ) J − + J − ( J z − S z ) ] | Σ = − , P, J , p〉 2 2 J −P+S−Σ+ p = −h2 (−1) 2 (P + 1) f (J, P) 1 1 〈Σ = + , − P, J , p | −2h2L -2 J z S − | Σ = + , P, J , p〉 = 2h2 (−1) J − P + S −Σ + p P 2 2

HSR,PERT ε1 1 1 〈Σ = − , −( P − 1), J , p | ε 1L -2 J + S + | Σ = − , P, J , p〉 = ε1 (−1) J − P + S −Σ + p f ( J , P − 1) 2 2

ε2a 1 1 〈Σ = + , − P, J , p | 2ε 2 aL -2 J z S − | Σ = + , P, J , p〉 = −2ε 2 a (−1) J − P + S −Σ + p P 2 2

ε2b 1 1 〈Σ = + , −( P + 1), J , p | 2ε 2bL -2 J − S z | Σ = − , P, J , p〉 = ε 2b (−1) J − P + S −Σ + p f ( J , P ) 2 2 _______________________________________________________________________ a J+|P,J〉=f(J,P-1)|P,J-1〉 and S+|S,Σ =-1/2〉=|S,Σ =+1/2〉. f (J, P) = J(J + 1) − P(P + 1) b

For these terms, it is somewhat misleading to add the term in L +2 . In Eq. (1), L -2 acts on the first term to change Λ from +1 to -1 and gives zero on the second; L +2 gives zero on the first term and changes Λ from -1 to +1. Thus the effect is to switch the first term to the second and the second to the first in addition to what the operators multiplying L do. One can determine the matrix element by consideration of the first term only.

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Table 2. Hamiltonian Matrix Elements of CHD2O or CH2DO in the Principal Axis System: a, b < −Σ, − P, J , p | H Q | Σ, P, J , p >= −(−1) J − P + S −Σ+ p

δE 2

1 asym < Σ, P + 2, J , p | H ROT | Σ, P, J , p >= − ( B − C ) ⋅ f ( J , P) f ( J , P + 1) 4 1 asym < Σ, P − 2, J , p | H ROT | Σ, P, J , p >= − ( B − C ) ⋅ f ( J , P − 1) f ( J , P − 2) 4 1 1 1 asym < Σ = , P − 1, J , p | H ROT | Σ = − , P, J , p >= ( B − C ) f ( J , P − 1) 2 2 2 1 1 1 asym < Σ = − , P + 1, J , p | H ROT | Σ = , P, J , p >= ( B − C ) f ( J , P) 2 2 2 asym < Σ, P + 1, J , p | H COR | Σ, P, J , p >= −Cζ t sin θ f ( J , P) asym < Σ, P − 1, J , p | H COR | Σ, P, J , p >= −Cζ t sin θ f ( J , P − 1) asym < −Σ, P, J , p | H COR | Σ, P, J , p >= Cζ t sin θ

1 aζ e d sin θ 2 1 asym | Σ, P, J , p >= ε ac P < −Σ, P, J , p | H SR 2 1 asym < Σ, P + 1, J , p | H SR | Σ, P, J , p >= ε ac f ( J , P )Σ 2 1 asym < Σ, P − 1, J , p | H SR | Σ, P, J , p >= ε ac f ( J , P − 1)Σ 2 1 1 1 asym < Σ = , P − 1, J , p | H SR | Σ = − , P, J , p >= (ε bb − ε cc ) f ( J , P − 1) 2 2 4 1 1 1 asym < Σ = − , P + 1, J , p | H SR | Σ = , P, J , p >= (ε bb − ε cc ) f ( J , P ) 2 4 2

asym < −Σ, P, J , p | H SO | Σ, P, J , p >=

a

. Only asymmetric terms of the Hamiltonian are given here. The symmetric terms can be

found in Ref (10), except that the spin-orbit and Coriolis elements found there should be multiplied by cosθ. b

The matrix elements here are for CHD2O. For CH2DO, b and c should be interchanged.

See text.

27

Fig.1. CH2DO (top) and CHD2O (bottom) in the principal (left) and internal (right) axis systems. θ is exaggerated.

28