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Journal of Molecular Spectroscopy 241 (2007) 15–17

Determination of the spin–spin centrifugal distortion parametersfor deuterated selenoformaldehyde in its first excited triplet state

Christian Hill, John M. Brown *

The Physical and Theoretical Chemistry Laboratory, Department of Chemistry, Oxford University, South Parks Road,

Oxford OX1 3QZ, England, UK

Received 18 October 2006Available online 10 November 2006

Abstract

Recently-formulated centrifugal distortion corrections to the spin–spin Hamiltonian have been used to analyse the first excited tripletstate (a3A2) of deuterated selenoformaldehyde. By including six spin–spin centrifugal distortion parameters, it is possible to accountfor the energy levels of this state using a single set of three rotational constants, A, B, and C, where previously nine spin-dependentparameters were required.� 2006 Elsevier Inc. All rights reserved.

1. Introduction

The deuterated selenoformaldehyde molecule, D2C80Se,is an orthorhombic asymmetric rotor belonging to the C2v

molecular point group. The high-resolution laser-inducedphosphorescence spectrum of the vibronically-induced 41

0

band of its ~a3A2 � ~X 1A1 electronic transition has beenrecorded by Joo et al. [1]. The a3A2 state is best describedby a case (ab)Ia coupling scheme [2,3] in which one spin-component (|zæ, corresponding to R = 0) is well separatedfrom the other two (|xæ, |yæ, corresponding to linear combi-nations of R = ±1); it lies 129 cm�1 above them. In termsof the spin–spin interaction constants a and b [4], this situ-ation arises when |a|� |b| and a < 0. In their paper, Jooet al. were able to fit the spectrum only by assuming differ-ent sets of rotational constants, AR, BR, and CR in each ofthe spin states. In this article, we demonstrate that the spec-trum can be satisfactorily fit to a single set of parameters byincluding the centrifugal distortion corrections to the spin–spin interaction. The Hamiltonian describing these terms isderived in a companion paper [5] to this one.

0022-2852/$ - see front matter � 2006 Elsevier Inc. All rights reserved.

doi:10.1016/j.jms.2006.10.011

* Corresponding author. Fax: +44 1865 275410.E-mail address: jmb@physchem.ox.ac.uk (J.M. Brown).

2. The Hamiltonian

The effective Hamiltonian used to describe deuteratedselenoformaldehyde in its a3A2 state is that derived withinthe Watson A reduction [6] for a Hund’s case (a) basisand may be divided into four terms

H eff ¼ H rr þ H cd þ H srcd þ H sscd: ð1Þ

Here, Hrr is the Hamiltonian for a rigid rotating moleculewith electron spin angular momentum and magneticinteractions

H rr ¼ �BðJ � SÞ2 þ ðA� �BÞðJ z � SzÞ2

þ B� C4

� �ðJþ � SþÞ2 þ ðJ� � S�Þ2h i

� a0ðJ � S � S2Þ þ a½�S2 � 3Sz � ðJ z � SzÞ þ J � S�

þ að3S2z � S2Þ þ b

2ðS2þ þ S2

� � JþSþ � J�S�Þ

þ b2ðS2þ þ S2

�Þ; ð2Þ

where �B ¼ ðBþ CÞ=2. The matrix elements for this part ofHeff are well-known and given elsewhere [7]. The centrifu-gal distortion Hamiltonian is

Table 1Fitted parameters (in cm�1) for the a3A2 state of deuteratedselenoformaldehydea

This work Joo et al. [1]

16 C. Hill, J.M. Brown / Journal of Molecular Spectroscopy 241 (2007) 15–17

H cd¼�DN ðJ�SÞ4�DNKðJ�SÞ2ðJ z�SzÞ2�DKðJ z�SzÞ4

�1

2dN ðJ�SÞ2þdKðJ z�SzÞ2;ðJþ�SþÞ2h

þðJ��S�Þ2iþþUKðJ z�SzÞ6; ð3Þ

and the spin–rotation centrifugal distortion Hamiltonian is

H srcd ¼DsKðJ�SÞ3SzþDs

KN ðJ z�SzÞ2ðJ�SÞ �Sþ1

2Ds

NK ½ðJ�SÞ2;ðJz�SzÞSz�þþDs

N ðJ�SÞ2½ðJ�SÞ �S�þdsN ½ðJþ �SþÞ2þðJ� �S�Þ2�½ðJ�SÞ �S�

þ1

2ds

K ½ðJþ �SþÞ2þðJ� �S�Þ2;ðJ z�SzÞSz�þþUs

KðJ z�SzÞ5Sz: ð4Þ

The matrix elements for Hrr + Hcd + Hsrcd are given inAppendix A of Judge et al. [3]; we give corrections to afew typographical errors in that paper in Appendix A ofthis work. The final term in Heff has not previously beenconsidered in detail. Its reduced form is

H sscd¼1

2Da

N ½N2ð3S2z �S2Þþð3S2

z �S2ÞN2��þDa

K ½N 2z ð3S2

z �S2Þþð3S2z �S2ÞN 2

z �þda

N ½ðN 2x�N 2

yÞð3S2z �S2Þþð3S2

z �S2ÞðN 2x�N 2

yÞ�þDb

N ½N2ðS2x�S2

yÞþðS2x�S2

yÞN2�þDb

K ½N 2z ðS2

x�S2yÞþðS2

x�S2yÞN 2

z �þdb

N ½ðN 2x�N 2

yÞðS2x�S2

yÞþðS2x�S2

yÞðN 2x�N 2

yÞ�g; ð5Þ

and its matrix elements in a Hund’s case (a) basis are givenin Appendix B of the companion paper [5].

Rotational constants

A 4.576766(34) 4.577054(11)b

B 0.3233725(16) 0.3237253(12)C 0.3033732(16) 0.3039124(12)

Centrifugal distortion constants

104DK 1.3670(48) 1.3546(26)106DNK 7.609(81) 7.774(03)107DN 2.717(11) 2.7893(24)105dK �1.068(70) 42.7(18)108dN 1.15(10) 1.655(23)108UK �2.10(29) �2.69(17)

Spin constantsa 0.44878(23) 0.44861(04)a0 0.43552(54) 0.46175(07)b �0.02128(28) �0.00294(04)a �42.22000(28) �42.22398(07)b 2.2373(24) 2.3009(05)

Spin–spin centrifugal distortion constants

3. Spectral fitting

The good quantum numbers in the Hund’s case (a) cou-pling scheme are:

J total angular momentum quantum number corre-sponding to J,

p (signed) component of J along the molecule-fixedz-axis,

M component of the total angular momentum alongthe space-fixed Z-axis,

S electron spin angular momentum quantum num-ber corresponding to S,

R component of the electron spin angular momen-tum along the molecule-fixed z-axis.

104DaN �3.544(24)

104DaK 1.71(13)

106daN �3.58(16)

104DbN �1.352(27)

DbK 0.0850(19)

104dbN 1.167(36)

T0 12373.44434(90) 12373.3842(5)

a Error limits are 1r, right justified to the last digit.b Rotational constants from Joo et al. are those for the R = 0 spin state.

Writing P = |p|, we construct the symmetrized (parity-adapted) case (a) basis, |wsæ|wræ, from the spin functions

jwsi ¼jxi ¼ 1ffiffi

2p jR ¼ þ1i � jR ¼ �1ið Þ;

jyi ¼ 1ffiffi2p jR ¼ þ1i þ jR ¼ �1ið Þ;

jzi ¼ jR ¼ 0i;

8><>: ð6Þ

and the rotational wavefunctions

jwri ¼

1ffiffi2p jJP oddi þ jJP oddið Þ;1ffiffi2p jJP oddi � jJP oddið Þ;1ffiffi2p jJP eveni þ jJP evenið Þ;1ffiffi2p jJP eveni � jJP evenið Þ:

8>>>>><>>>>>:

ð7Þ

This factorizes the Hamiltonian matrix of given J intoblock-diagonal form as described in ref [3].

The energy levels of the a3A2 state were calculated fromthe data of Joo et al. [1] by subtracting their known lowerstate energy levels [8] from the transition wavenumbers.These upper-state levels (for rotational quantum numberJ up to 50) were fitted using a linear-least squares algo-rithm. As in ref [1], at each iteration of the algorithm datapoints with a residual greater than 3r were excluded fromthe fit, but re-introduced on subsequent iterations if theirresiduals fell below that threshold. Some of the energy lev-els are severly perturbed, leading to problems in assigningquantum numbers to them; rather than following the some-what complex procedure suggested by Judge et al. [9] forsuch levels, we exclude from the fit energy levels which can-not be unambiguously assigned. These comprise roughly5% of the data and so our data set of 1379 levels is corre-spondingly slightly smaller than the 1455 derived from the4208 transitions considered by Joo et al. [1]. The overall fitstandard deviation was 0.0025 cm�1, comparable with thereported experimental accuracy, and close to the quality

C. Hill, J.M. Brown / Journal of Molecular Spectroscopy 241 (2007) 15–17 17

of the fit by Joo et al. (0.0024 cm�1). The fitted parametersare given in Table 1.

4. Further applications of Hsscd

We have been able to describe the energy levels of thefirst excited triplet state (a3A2) of deuterated selenoformal-dehyde with a single set of rotational constants, A, B, andC. This makes it possible to determine the molecular geom-etry of this electronic state directly once the effect of the var-ious transformations [4,5,10] on the Hamiltonian has beenunravelled, something that is not feasible from the ninespin-dependent parameters deduced by Joo et al. [1]. Unfor-tunately, information on at least one other isotopic form ofselenoformaldehyde is required for a full r0 structure deter-mination. Using the rotational constants in Table 1 for thea3A2 state of D2CSe, the inertial defect is determined to be�0.26817(39) amu A2; this is large and negative, consistentwith the non-planar structure of the molecule in this state.

The spin–spin centrifugal distortion parameters havenot been included in existing spectral analysis studies andit seems likely that the Hsscd Hamiltonian used here wouldfind application in the analysis of other orthorhombicasymmetric tops in electronic states with S P 1. In partic-ular, transitions within the triplet ground states of CH2 andCD2 have been measured in the sub-millimetre wave region[11,12], but so far only the Da

K parameter has been deter-mined for these molecules.

Appendix A. Errata for the matrix elements ofHrr + Hcd + Hsrcd given in Ref. [3]

A few typographical errors appear in both appendices toJudge et al.’s paper [3] giving matrix elements for theHrr + Hcd + Hsrcd operator. We give corrections here,using the notation of that paper. From Appendix A of[3]: in the matrix element Æp � 1,R = 0|H|p,R = �1æreferred to as H 0�1

p�1p, the term in dN ought to be

� dN 0:5f þðpÞsþð�1Þ½�4JðJ þ 1Þ � 10� fþðp � 1Þ2

� f�ðpÞ2 � 4p�; ðA1Þ

in the matrix element Æp � 1,R = 1|H|p, R = 0æ referred toas H 10

p�1p, the term in dN ought to be

� dN

ffiffiffi2p

2fþðpÞ½�4JðJ þ 1Þ � 10� f 2

þðp � 1Þ � f 2�ðpÞ

þ 4ðp � 1Þ�: ðA2Þ

From Appendix B of [3], the correct matrix elements inthe example symmetrized EPLUS matrix are

P 0 ¼P�1 : ZX p;p0 ¼1ffiffiffi2p ½H 01

pp0 �H 0�1pp0 þH 0�1

p�p0 �; ðA3Þ

P 0 ¼Pþ1 : ZY p;p0 ¼1ffiffiffi2p ½H 01

pp0 þH 0�1pp0 �; ðA4Þ

P 0 ¼P ¼1 : X 11¼�1

2½H 11

11�2H 1�111 �2H 11

1�1þH 1�11�1þH�1�1

11 �;

ðA5Þ

P 0 ¼P ¼1 : Y 11¼1

2½H 11

11þ2H 1�111 þ2H 11

1�1þH 1�11�1þH�1�1

11 �;

ðA6Þ

P 0 ¼P ¼1 : XY 11¼�1

2½H 11

11þH 1�11�1�H�1�1

11 �H�111�1�; ðA7Þ

P 0 ¼P : XY pp¼�1

2½H 11

pp�H�1�1pp �: ðA8Þ

References

[1] D.-L. Joo, D.J. Clouthier, R.H. Judge, J. Chem. Phys. 112 (2000)2285–2291.

[2] F. Creutzberg, J.T. Hougen, Can. J. Phys. 45 (1967) 1363.[3] R.H. Judge, A.A. Korale, J.J. York, D. Joo, D.J. Clouthier, D.C.

Moule, J. Chem. Phys. 103 (1995) 5343–5356.[4] J.H. Van Vleck, Rev. Mod. Phys. 23 (1951) 213–227.[5] C. Hill, J.M. Brown., J. Mol. Spectrosc. 104 (1996) 2167–2171.[6] J.K.G. Watson, in: J.R. Durig (Ed.), Vibrational Spectra and

Structure, 6, Elsevier, 1977, pp. 1–89, Chapter 1.[7] C. Di Lauro, J. Mol. Spectrosc. 35 (1970) 461.[8] D.-L. Joo, D.J. Clouthier, R.H. Judge, D.C. Moule, J. Chem. Phys.

102 (1995) 7351.[9] R.H. Judge, E.D. Womeldorf, R.A. Morris, D.E. Shimp, D.J.

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[11] H. Ozeki, S. Saito, Astrophys. J. Lett. 451 (1995) L97–L99.[12] H. Ozeki, S. Saito, J. Chem. Phys. 104 (1996) 2167–2171.