The calculation of the energy levels of an asymmetric top free radical in a magnetic field

THE CA~C~LAT~N OF THE ENERGY LEVELS OF AN ASYMMETRIC TOP FREE RADICAL IN A MAGNETIC FIELD

NORTH-HOLLAND

- AMSTERDAM

T.J. Sears / Energy levels of an asymmetric top free radical

Contents

1. Introduction

.......................................

2. The effective Hamiltonian

..............................

3. Basis set and matrix elements 4. Machine implementation.

...........................

..............................

4 5

12 21

5. Description of package ................................

25

6. Analysis of the millimeter and submillimeter wave spectrum of the HCO radical .......................................

27

7. Summary ..........................................

30

References ...........................................

31

Computer Physics Reports 2 (1984) 1-32 North-Holland, Amsterdam

THE CALCULATION OF THE ENERGY LEVELS OF AN ASYMMETRIC RADICAL IN A MAGNETIC FIELD Trevor

J. SEARS

Department of Chemistry, Brookhaven National Laboratory, Received

TOP FREE

Upton, NY 11973, USA

15 March 1984

The form of the effective Hamiltonian operator for an asymmetric top free radical subject to a magnetic field is presented with an emphasis on machine calculation of the eigenvalues of the operator. Problems associated with the choice of a suitable basis set are discussed and the matrix elements of the Hamiltonian are given in a fully coupled parity conserving basis. The calculation of frequencies, relative intensities and Zeeman tuning rates of gas phase magnetic resonance transitions are illustrated using the available high resolution data for the formyl radical as an example.

0167-7977/84/$09.60 0 Elsevier Science Publishers (North-Holland Physics Publishing Division)

B.V.

4

T.J. Sears / Energy Ievels of an asymmetric top free radical

1. Introduction This review describes the theory behind a Fortran IV package developed over a period of several years for the analysis of high resolution spectra of asymmetric top molecules with non-zero resultant electron and nuclear spin angular momenta. A particular attribute of this package is the comprehensive treatment of the Zeeman effect in such molecules that derives from the author’s involvement in various types of gas phase magnetic resonance spectroscopy. In this experiment, a fixed frequency source of radiation is used which is close to, but not coincident with, an allowed transition frequency in the molecule of interest. Components of the molecular transition are then tuned into resonance with the radiation source by the application of a magnetic field and detected by absorption of the radiation. The experiment is therefore only sensitive to paramagnetic species such as molecular free radicals. Initially, experiments were performed at microwave frequencies using a klystron as the radiation source [l], however, since the source need not be continuously tunable, use has recently been made of the large number of line tunable gas lasers currently available. The high power and monochromicity of these lasers contribute to the high sensitivity of the technique. The need for the type of calculation discussed in this article has increased over the past decade as the new, laser based, variants of the magnetic resonance technique have enabled detailed study of molecular free radicals which have long been recognized as important intermediates in many chemical reactions. An early example of the success of laser magnetic resonance spectroscopy was the detection of the hydroperoxyl radical [2], HO,, and the later detailed analysis [3,4] and interpretation of the spectra. Subsequently, many species have been studied using similar techniques [5,6] and there is good reason to believe that as experimental methods become further refined, the list will continue to expand in both the number and type of molecule studied. Analysis of the kind of high resolution spectra considered here can be broadly divided into three stages. Firstly, it is necessary to make the quantum number assignments to the observed spectral lines. For magnetic resonance spectra, this process is often not trivial since the experiment depends upon tuning components of molecular transitions into coincidence with a fixed frequency source. The spectra are obtained at a series of separate frequencies and it is sometimes difficult to make consistent unambiguous assignments. In most molecules studied to date, use has been made of the results of earlier lower resolution work in order to make the initial assignments. Having arrived at an assignment of the spectra, the second stage in the analysis is to refine the model representing the molecule, usually the effective molecular Hamiltonian operator, by direct comparison of observed and calculated spectra. Finally, the derived model parameters should be interpreted in terms of the detailed molecular structure; the intramolecular potential function and spin densities for example. In what follows we look at the construction of a suitable molecular model which is designed to aid in the type of analysis outlined above. Throughout the review, an attempt has been made to use a consistent notation and to include all the relevant equations for machine implementation. In the next section, the form of the effective Hamiltonian for an asymmetric top molecule is discussed and the important results are quoted. Some discussion of the reasons for adopting particular forms of the operators in a machine calculation is included. Section 3 discusses the choice of a suitable basis set for the calculation and shows how the basis set for any particular calculation can be truncated to the minimum size consistent with the accuracy required for the

T.J. Sears / Energy levels of an asymmetric top free radical

5

problem in hand. The matrix elements of the various terms in the effective Hamiltonian are quoted in the basis set described and the rationale for choosing the forms adopted in the program discussed. In section 4, calculation of transition frequencies and intensities from the diagonalized Hamiltonian matrix is outlined. Problems associated with the identification of particular eigenstates are discussed and the advantages and disadvantages of various methods compared. We also consider the computation of the partial derivatives required in the least squares refinement of the parameters in the effective Hamiltonian and describe how this is achieved in practice. The final section contains a detailed description of the various modes of operation of the spectrum analysis package and an example of its use in a real problem. In a separate article *, the structure of a Fortran IV program based on the model discussed in this review is described in more detail together with the required dataset structure and several examples of its use.

2. The effective Hamiltonian In this section we consider in detail the form of the ,molecular model that we use to interpret high resolution spectra of asymmetric top free radicals. The molecular energy levels are the eigenvalues of the effective Hamiltonian operator for the molecule in the particular vibronic state of interest. The derivation of a soundly-based effective Hamiltonian is beyond the scope of this article and has been discussed in the literature [7-131; we consider here only the form of the operators involved. An effective Hamiltonian is defined such that its matrix elements span only a single vibronic state. The effects of matrix elements connecting different vibrational and electronic manifolds have been collapsed into effective molecular parameters for the state in question. Such a procedure naturally has the effect of reducing the size of the problem; it also removes explicit references to vibronic states which may at best be only poorly characterized. In this way, after fitting an observed spectrum, we are free to interpret the derived parameters for the state in terms of admixtures of other vibronic states in several different ways in order to determine which best fits the observations. In short, a soundly-based effective Hamiltonian separates the choice of a specific model from the mechanical fitting of the spectrum. The complete operator is assumed to consist of a sum of terms, each identified with a specific contribution to the molecule’s energy. Thus, we write H EFF=HR+HcD+HFs+HHFs+HQ+Hz. refer to the rotational energy and its centrifugal distortion correction, Here, H, and H,, respectively. HFs takes account of the fine structure energy contributions; for a doublet state, this will consist of just the spin-rotation interaction HSR and HsRCD. For states of higher multiplicity we must take account of the electron spin-spin dipolar interaction H,, also. HHFs represents the electron-spin, nuclear-spin hyperfine interaction and HQ represents the quadrupole interaction for a nucleus with spin equal to one or more. Finally, Hz represents the interaction between the molecule and an external magnetic field. These terms are sufficient to understand and analyze all but the highest resolution spectroscopic data, the energies calculated from (1) should be accurate to at least 0.1 MHz. * Published in Computer Physics Communications

[51].

T.J. Sears / Energy levels of an asymmetric top free radical

6

We now consider the explicit forms of the individual operators in eq. (1). For machine computation, the formulation of the effective Hamiltonian in terms of irreducible tensor operators as described by Bowater, Brown and Carrington [13] has several advantages. The primary one is that the matrix elements are easily calculated in any chosen basis set by using algebraic methods [14,15] and the resulting general expressions which involve vector coupling coefficients are easily coded and checked. These advantages are particularly important when the operator involves several coupled angular momenta. In the case of simpler operators, for example, those representing the rotational motion, there is a good argument in terms of efficiency for retaining the well known and simple algebraic expressions, these may be evaluated from the general expressions given here, since the matrix elements are few in number and easily checked. In the following discussion we will present the operators in their irreducible tensor form and, where it is useful, refer these to the more familiar Cartesian forms. We take the rotational Hamiltonian in irreducible tensor form as [13] H, = i

Z+(B)*Z+(iV,

N).

(2)

k=O

Here, Tk( the tensor rotational rotational

B) is an irreducible tensor of rank k and Tk( N, N) is one of the same rank formed by product [14,15] of two first rank irreducible tensor operators, T’(N), representing the angular momentum apart from electron spin. With the definition of the effective Hamiltonian as given in eq. (2) and the expression

H, (Cartesian)

= AN: + BN: + CN:

(3)

for the same operator in the more familiar Cartesian components .Tqk(B) to the usual rotational constants T,‘(B)=-$A+B+C),

tensor

T;(B)=+(2A-B-C),

representation

we can relate

T:,(B)=+(B-C).

the

(4

The first rank (k = 1) terms are zero in the principal axis system of the inertial tensor. Throughout this section, we implicitly assume that the constants appearing in the effective Hamiltonian refer only to the relevant vibronic state. In eq. (3), N,, NY and N, are the components of the rotational angular momentum operator N referred to the molecule fixed x, y and z axes. We have assumed an I’ representation [ll] (x ++ b, y +-+c, z ++ a) appropriate to a near prolate top asymmetric rotor. The centrifugal distortion Hamiltonian has been described in detail by Watson [ll] and we assume the, Cartesian, form H,,

= -A,N;

- A,N=N;

+@ K N6+@ I +

KN

- A,N4

- 2S,N2(

N: + NT) - S,[ N: + N! , N;],

N2N4z

QNKN4N,2 + @,N6 + 2+,N4(

N: + NT)

(5)

T.J. Sears / Energy levels of an asymmetric top free radical

7

where N, = (iV, f ilv,)/fi and the A and @ constants are the effective quartic and sextic centrifugal distortion parameters. One octic centrifugal distortion parameter L, has also been included; this is expected to be the largest octic distortion parameter. L, and other higher order distortion parameters are expected to have significant effects on the rotational energies only for light molecules and for those exhibiting large amplitude vibrational motions. Eq. (5) corresponds to Watson’s ‘A’ reduction [ll] of the centrifugal distortion Hamiltonian and is important because the non-zero matrix elements of the operator are simple and connect only states of the same K, where K = (N,), and those differing in K by two in all orders. The simplicity of the matrix representation of the operator in this form makes it especially suited to machine calculations and for this reason eq. (5) represents the most commonly used form of the operator. The ‘A’ reduction does, however, suffer from the disadvantage that it is not well defined when the molecule is a prolate or near prolate top. In such cases, it is preferable to use the ‘S’ reduced form [ll] which remains well defined under all circumstances, and on these grounds alone, the general adoption of the ‘S’ reduced operator would appear to be preferable [11,16]. In practice, situations where the ‘A’ reduced operator has failed have been very rare although in the few examples where direct numerical comparisons have been made [16], the ‘S’ reduced operator produced higher quality fits to experimental data with less strongly correlated parameters than did the ‘A’ reduced operator when the molecule in question was a near prolate symmetric top. The disadvantage of the ‘S’ reduced formulation is that the matrix elements of the operator are less convenient to program and that a large body of experimental data exists for which fits to the ‘A’ reduced operator have been performed. In this review the ‘A’ reduced form of the centrifugal distortion and quartic spin rotation Hamiltonians are discussed in detail. Versions of the program exist using both the ‘A’ and ‘S’ reduced operators; however, popular usage of the ‘A’ reduced form of the Hamiltonian seems likely to continue despite the advantages inherent in the alternative form. The reduction of the centrifugal distortion Hamiltonian containing all the symmetry allowed terms to a form equivalent to eq. (5) has not been performed in an irreducible tensor formulation and this is the reason for adopting the Cartesian representation for the centrifugal distortion operator and also for the quartic spin-rotation operator discussed below. Classically, the electron spin-rotation interaction arises from the coupling between the spin dipole moment and the magnetic field created by the charged particles in the molecule as they rotate. The form of the effective spin-rotation Hamiltonian has been derived by Van Vleck [7] and Curl [8] from a consideration of mixing of other electronic states by the combined effects of the spin-orbit coupling and the Coriolis term in the rotational Hamiltonian. Subsequently [17], it was discovered that in order to understand the details of the spectra of small asymmetric top free radicals, it was necessary to allow for centrifugal distortion contributions to the spin-rotation energy. These arise [12] in a completely analogous way to the more familiar pure rotational interaction. Brown and Sears [18] have shown that similar indeterminacies to those recognized by Watson [ll] in the centrifugal distortion correction to the rotational energy occur for the spin-rotation interaction also. After Brown and Sears [18], we adopt the following forms for the operators:

(6)

T.J. Sears / Energy ieueis of an asymmetric top free radical

8

and

Again, we have chosen to assume an I’ identification of the molecule fixed axes. In contrast to the reduction of the centrifugal distortion Hamiltonian [ll], the number of determinable parameters in the effective spin-rotation Hamiltonian depends on the symmetry of the molecule. In eq. (6) the last two terms are zero for any molecule other than one with no elements of symmetry (belonging to point group C,). Additionally, the fourth term is zero for molecules belonging to orthorhombic point groups. At the present time, there are no high resolution data available for molecules belonging to the C, point group and the detailed effects of the terms involving Z,, and cbc have not been investigated. Bowater, Brown and Carrington [13] considered the irreducible tensor formulation equivalent to eq. (6). In this representation HSR may be written

H,,=$

&

(Tk(e)*Tk(N,

S)+

Tk(N,

k=O

S)*Tk(e)].

(8)

The restricted sum is over even k only and corresponds to the reduction given in eq. (6). The relationships between the eaB parameters in eq. (6) and the components of the irreducible tensor Tk(e), are

(9)

The centrifugal distortion correction represented by eq. (7) is cast in the ‘A’ reduced form for the same reasons as we adopted the analogous form of the rotational centrifugal distortion correction. Eq. (7) is strictly only appropriate for asymmetric top molecules belonging to orthorhombic point groups. There are additional terms in the operator when the molecule has fewer elements of symmetry however these have yet to be explicitly calculated and, to date, eq. (7) has proved to be quite adequate in representing the operator for less symmetrical species. The details of the irreducible tensor form of HsRCDhave not yet been considered. Note that we have included one term belonging to the sextic spin-rotation Hamiltonian. This has recently been found to be necessary to satisfactorily account for the spin-rotation splittings in HCO [20]. The last contribution to the fine structure Hamiltonian occurs only for molecules in electronic states of triplet or higher multiplicity and allows for the electron spin-spin dipole-dipole interaction between the unpaired electrons. In the irreducible tensor formulation we have [19]

H,, =

&-g2p;c T’(C)-T2(s;, Sj)’

(10)

i>j

where g is the spin g-factor, pB the Bohr magneton and the components of the second rank tensor

T.J. Sears / Energy levels of an asymmetric top free radical

T2( C) are related to the averages of the spherical harmonics parameters Ti( C) describe the spin-spin interaction and T,2(c)=

C (($q)“2y2,(s,

over the vibronic

$)/‘i;)*

9

coordinates.

The

(11)

irj

and + are the spherical polar angles defining the distance between them. This is equivalent ~7,211

8

H&Cartesian)

the relative positions of electrons i and j and rij is to the effective operator in Cartesian coordinates

= D(2S,’ - S,’ - Sy2)/3 + E( S,’ - SyZ),

(12)

with 20 = -3g2&T02(C) and 2E = -(6)‘/2g2&Ti2(C). Contributions to the effective parameters D and E derive from both first order spin-dipole coupling and from higher order spin-orbit mixing of other electronic states. Additionally, the reduction procedure used in deriving the effective spin-rotation operator (6) and (7) generates contributions [18] which have not been investigated in detail. The magnetic hyperfine Hamiltonian for a single nucleus with spin I may be written [13] H HFS=ar’(l)*T’(S)-(10)“2T*(I)*T1(S,

c’).

Here, the first term represents the isotropic, Fermi and the scalar product of the first rank irreducible and electron spin angular momenta. The second interaction energy and involves the tensor product defined by

T,‘(s, c’> = -m-)pm3gNPNc

(13)

contact interaction involving the parameter a tensor operators representing the nuclear spin term represents the spin-spin dipole-dipole T’( S, C2) whose space fixed components are

~~~)T,:w(;l;2 ‘J.

04)

PIP2

where pN is the nuclear magneton and g, the appropriate nuclear g-factor. The components Tq2(C) are defined analogously to those of the electron spin-spin dipolar interaction above

T,2(a = ((W’y2,(& 9)/R3). In Cartesian

representation

HHFs(Cartesian)

the hyperfine

= aS I + S l

l

T I. l

(15) interaction

may be written 06)

The relationship between the components of the traceless Cartesian dipolar hyperfine interaction tensor T in the molecule fixed axis system and the components Tq2(C) are given by Bowater, Brown and Carrington [13]. They are

10

T.J. Sears / Energy levels of an asymmetric top free radical

The Kj components are related to the (ij), parameters of Curl and Kinsey [22] by qj = (a + (ij),)/ 6. In general there are five independent components of T2( C) although symmetry restrictions may reduce this number in some cases. In the above discussion we have implicitly assumed that only one nucleus has a non-zero spin. The extension of eq. (13) to more than one nucleus is straightforward in principle since we merely have to add further operator terms of the form given by eq. (13) for each nucleus with non-zero spin. For the case of two such nuclei an appropriate basis set is ]~NKsJI,I,IF) where 1= Ii + I, and F = J + I. There have only been a few cases where this has been necessary, the most notable being in the analysis of high resolution microwave optical double resonance spectra of NH, by Hills et al. [23]. In the present paper we restrict the discussion to the case of single nucleus with non-zero spin. For molecules containing a nucleus with spin greater than l/2 we must also allow for the nuclear electric quadrupole interaction. In the irreducible tensor formulation this operator is given by [13] Ho=eT2(Q)*T2(vE).

(18)

Here -e is the electronic charge, the expectation value of T:(Q) is related to the nuclear quadrupole moment and the molecule fixed components of Tt( vE) are the components of the electric field gradient at the nucleus. The Cartesian form of the operator may be taken as [22] Ho (Cartesian)

= I

l

Q 1

(19)

l

and the components of the traceless quadrupolar interaction tensor Q in the molecule system are related to the components of the irreducible tensor T2( vE) by

fixed axis

(20)

with Q = 2(1, Finally, we molecule with Hz. Bowater, tensor notation

IlT,*(Q)l1, 1). wish to include terms in the Hamiltonian to take account of the interaction of the an external magnetic field. In eq. (1) this is represented by the Zeeman operator, Brown and Carrington [13] have given the form of this operator in irreducible as

k=O

Here the magnetic field direction first term in eq. (21) represents

defines the space fixed 2 axis and B, is the flux density. The the isotropic spin Zeeman interaction, the second term the

T.J. Sears / Energy levels of an asymmetric top free radical

11

anisotropic contributions arising from spin-orbit mixing of other vibronic states, the third term represents the interaction between the molecule’s rotational angular momentum and the external field while the last term represents the nuclear spin Zeeman interaction. The components T,k( gi) and T,k( g,) in the molecule fixed axis system are related to the more familiar Cartesian tensor components gij and gfj in the Cartesian form of the operator, H,(Cartesian)

= gs~s&SO

- p@*g,*S

+ pBB*gr*N-

gNpNBOIO.

(22)

The relationships between the irreducible tensor components T,k( gl) and the Cartesian anisotropic g;j tensor components are completely analogous to those of the spin-rotation interaction operator eq. (9). Similarly the diagonal components of the Cartesian rotational g-tensor gij defined by Flygare and Benson [24] are related to their irreducible tensor analogues by

-+Y+s:‘+P:‘L = &(2g:’ - g:” - g,yy), &(g,)

GYd= md

(23) = f(s:” -sryy).

The effects of off-diagonal components of the gf’ tensor on the molecular energy levels are small and have yet to be experimentally observed, however, those of the anisotropic spin g-tensor, gfj, can be important under some circumstances, specifically when the corresponding spin-rotation tensor elements Zij are important. Such effects have been observed in the far infrared spectrum of the HCO radical [25]. The form of the effective Hamiltonian considered here is derived on the assumption of a Born-Oppenheimer separation of nuclear and electronic motion. The effects of other electronic states are treated to second order in perturbation theory and additional contributions to the effective Hamiltonian can be expected in cases where there are close lying vibronic states. This situation is unlikely to occur when the electronic state in question is the lowest (ground state). However, as more detailed spectroscopic data become available for molecular free radicals in excited electronic states, there is a much greater chance of a breakdown in the perturbation theory approach used in the derivation of the effective Hamiltonian discussed here. Recent high resolution spectra of formaldehyde and thioformaldehyde [26-281 show that the first excited singlet and lowest triplet states of these molecules exhibit many local mutual perturbations. In this case, the effective Hamiltonian for a single state has to be modified and explicit reference included to the perturbing state or states. A similar, and simpler, situation occurs in some levels in the lowest singlet state of methylene [29] which are quite strongly perturbed by levels in excited vibrational states of the ground triplet electronic state. A more common shortcoming in the operator described here occurs in the form of the effective rotational and centrifugal distortion Hamiltonians. These are derived on the assumption that the vibrational potentials are reasonably well approximated by harmonic oscillator functions. When this is not the case, for example, in a molecule subject to large amplitude bending motion, the centrifugal distortion Hamiltonian does not converge very quickly. An extreme example of this type of behaviour is afforded by CH, in its ground electronic state [30-321. In this case, the large amplitude of the anharmonic motion in the bending fundamental results in large values and poor

12

T.J. Sears / Energy levels of an asymmetric top free radical

convergence of the centrifugal distortion Hamiltonian, Hc,. The practical results are that many distortion constants are required to fit few rotational intervals and that it is difficult to extract an accurate structure or bending potential function from the derived parameters. Also, predictions of molecular transition frequencies involving levels outside those explicitly included in the data are subject to large and indeterminate errors. These problems in z3B, CH, occur because of the vibrational bending motion is highly correlated with the rotational motion about the u inertial axis in CH,. The problem is better approached using a model such as the non-rigid bender (NRB) [32-341 which allows for explicit vibration-rotation coupling of this sort. However, the inclusion of fine and hyperfine interactions in the NRB model has yet to be performed. Additional complications arise when the effective Hamiltonian refers to a vibrational level which is perturbed by a close lying vibrational manifold through Coriolis or Fermi-type resonance terms. In high resolution spectra of asymmetric top free radicals such a situation has recently been observed in the ground electronic states of several molecules, notably DO, [35] and DC0 [36] where the two lower frequency fundamentals interact through a Coriolis mechanism. In these cases, a model using an effective Hamiltonian for a single vibronic state breaks down and the calculation of reliable energy levels requires explicit consideration of both of the states involved.

3. Basis set and matrix elements In order to calculate the molecular energy levels we have to set up the matrix of the effective Hamiltonian described in the previous section in a suitable set of basis functions and diagonalize it using a computer. In this section, we describe the form of the basis set that we adopt and the occasions for choosing one particular basis over any other. We then quote the matrix elements of the various terms in the Hamiltonian eq. (1) in a form suitable for machine calculation, and discuss how the size of the matrix that we have to deal with may be diminished by the application of symmetry restrictions and by basis set truncation based on considerations of acceptable accuracy. Before we can evaluate the matrix elements of the Hamiltonian, we must make some choices regarding the basis representation that we are to use. The correct choice is important because we wish to attach meaningful quantum labels to the eigenstates and connected to this we want to minimize the effects of off diagonal elements and hence the size of the matrix that we are required to set up and diagonalize. The first decision to make is on the type of angular momentum coupling scheme which is appropriate. For asymmetric top molecules, we expect the spin-orbit coupling to be very small and a coupling scheme that is analogous to Hund’s case (b) coupling [37] in diatomic or linear molecules is indicated. We may then choose to consider the nuclear spin angular momentum to be coupled to or uncoupled from the rotational angular momentum. Matrix elements of the various terms in (1) are available in the literature in both the fully coupled [13] N+S=J,

J+I=F

and the I-decoupled [12] representations. true situation when there is no external

(24) The former is expected magnetic field. However,

to more closely represent the as the applied magnetic field

T.J. Sears / Energy levels of an asymmetric top free radical

13

strength increases, the nuclear spin and electron spin tend to decouple from the molecular framework until the molecular eigenstates are best represented by a basis set in which the nuclear and electron spins are preferentially aligned with the external field. The magnetic field strength at which these processes occur depends on the size of the hyperfine and fine structure coupling as represented by the terms in the Hamiltonian. It would therefore seem that the optimum choice of basis set would depend on the exact experimental conditions for which the energy levels are required. To a certain extent this is true, however the differences in time required for machine construction and diagonalization of the Hamiltonian matrix is only very slightly dependent on the basis representation and it is preferable to choose one in which the matrix elements are the simplest and there is a smaller possibility of generating errors in the program code. For this reason, we will use a basis set represented by eq. (24), and possible problems associated with labelling the eigenstates that arise are discussed below. The rotational basis functions are chosen to be the symmetric top eigenfunctions which can be related to the elements of the Nth rank rotation matrix @$)(o)* by [14,15]

where w stands for the three Euler angles and in an I’ representation k = K, the component of N along the a molecular axis. The phase choice implicit in the matrix elements quoted below is that of Edmonds [15], matrix elements quoted by Van Vleck [7] and Raynes [9] differ from those quoted here by a factor of (- 1) in the ones off diagonal in N by one. We may represent our general basis vector by the ket

where r represents all of the other quantum numbers, such as vibrational and electronic state labels, needed specifying the state uniquely. In this basis set the matrix elements of the terms in the Hamiltonian (1) are given below. For the rotational Hamiltonian [13]

(27) and (NIIP(N,

N)((N) = (-)“f2k+

11‘hqN+

1)(2iv+

l,{F

;”

I&}.

(28)

The only non-zero matrix elements are those with AK = 0, + 2; they may be explicitly calculated by evaluation of the 6-j and 3-j symbols. The non-zero matrix elements of the centrifugal distortion operator are [ 111,

14

T.J. Sears / Energy levels of an asymmetric top free radical

(NKsJIFM,IH,,INKsJIFM,)

= -A,K4

- A,K2N(

N + 1) - A,( N( N + 1))2 + @,K6

- L,K’,

(29)

xf(N, K) + +,(NtN + 1)j2f(N,K),

(30)

where f(iv,

+K)=[(N~K)(N*K+~)(N+K+~)(N~K+~)]’/~.

(31)

Matrix elements of the spin-rotation terms as represented by (6) and (8) have been given in the literature [13,18]. The general expression in the irreducible tensor formulation is [13]

x [(2N + 1)(2N’

+ l)]“‘(

-y+,,+, 1

;

xf[(-)*[N(N+1)(2N+l)]‘/‘(:,

+ [iv’(iv’+

x C( - y4

1)(2iv’+

i

“‘,,

;,

;

l)j’/2(

f

ly,

; “Kqw.

;)

;)

;,)I

(32)

i

The general expression for the matrix elements of the spin-rotation operator in eq. (32) is identical to that given by Bowater, Brown and Carrington [13] except for the restricted sum over k noted in the previous section. It may be used to calculate explicit matrix elements which agree with those of Raynes [9] except for a phase factor of ( - 1) in those with N’ = N + 1. The matrix elements of the quartic spin-rotation Hamiltonian, HsRCD, have been given by Brown and Sears

T.J. Sears / Energy levels of an asymmetrictopfree radical

15

[18]. Explicitly they are

( NKsJIFM,l J&la INKSJIFM~) = -~A~K4+~~K6+(A~~+A~~)K2N(N~l) +A~(N(N+ (NK+2SJIFM#IsRcD

l))2]~(N~,

INKSJIFM,) = -(+a;[K2+(K+2)2j

(33) +S&N(N+l))

x C(N)J’(N, K),

(N-Xi-

(34

2SJIFi&(HsRCDINKSJIFM~) = +$(G$/'N){K(N+K) +(Kt2)(N,K+2)}[(N+K-1)

where the quantities B(N)= F(N,

-C(N),@N(N+l)],

C(N)=J(J.+l)-N(N+l)-S(S+l),

~K)=[(N~K)(N~K+~)(N~K-~)(N~K~~)]~'~, P(N)Q(N-1) (2N-1)(2N+l)

P(N)=(N-J+S)(N+J+S+l),

(37)

1’2 I

’

Q(N)=(S+J-N)(N+J-S+l),

have been defined following Raynes [9]. The matrix elements of the spin-spin dipolar interaction are given in general by

{N'K'SJI~~~l~~slNK~JIF~~)= -6"2gzp;(-)N+S+' ;

x[(2N'+1)(2N+

1)]1’2

;

;')zT,2(C)(-)N'-K' 4 ;

(38) The reduced matrix element may be evaluated to give

16

T.J. Sears / Energy levels of an asymmetric top free radical

C (SI(T.2(S;Y Sj)[lS) = C (S,~[T’(si)/[S,)(S,/~T.‘(S,)(~Sj)(5)”2 I>j i>j

i

s;

s;

1

Sj

Sj

1 (2S+

s

s

2i

Evaluation of these equations for the specific case i = 2, j = 1, si = sj = l/2 expression quoted by Hallin et al. [19]. The matrix elements of the magnetic hyperfine interaction are [13] (N’K’SJ’IFM,IH,,,INKSJIFM,)

= ( -),+I+‘[ x (27

;’

(39)

and S = 1 gives the

S( S + 1)(2S + 1)1( I + 1)(21+

+ 1)(2J + 1)]1’2(:

1).

1)

;}

a8 NN’ S KK’

-(30)1’2g~Bg,~,[(2N’

N’

X

The matrix elements

-K’

of the electric quadrupole

( N ‘K ‘SJ’IFM,I HQ\ NKSJIFM,)

(_;

= -$

2

N

q

K

interaction

+ 1)(2N + I)]“*

(40)

are [13]

;)-‘(-,J+~+F

0’

I

J’

i JI

F 2

‘I

N’+S+J

X [ (2 J’ + 1)(2 J + l)]“*

2N’ + 1)(2N + l)]“*

xC(-lN’-K’ _$ ; ; 4

i

T(f(vE), i

(41)

T.J. Sears / Energy Ievels of an asymmetric top free radical

where the electric quadrupole moment Q is defined elements of the Zeeman operator are given by [13]

in the previous

17

section.

(_)N+S+J’+lga

;

(-)“(f(2k+

the matrix

6 s

-

Finally,

N’N

K’K

1)]1’2

k=O

gl”, S)~~NKSJ)

x (N’K’SJ’I)T.‘(

J’ N

+(

-)N’+S+J+l

f)‘(

2 J’ + 1)(2 J + 1)]1’2

I

x (N’K’IlT.‘(g,“,N)IlNW -gNp.N(SN’NSK’KSJ’J(

x [1(1+

1)(21+

and the reduced matrix elements are as defined in ref. [13]. In order to calculate the molecular energy levels, we must the general expressions for the matrix elements given above the appropriate similarity transformation. Since treatment hand is impractical, this process is now normally carried

1

-)J+‘+F’+l ;

;’

:)

1

(42)

set up the and bring of all but out using

Hamiltonian matrix using it to diagonal form using the simplest problems by a computer. In principle,

l)]“’

T.J. Sears / Energy levels of an asymmetric top free radical

18

however, the full matrix including the matrix elements of the Zeeman operator is infinite, because of the presence of matrix elements off diagonal in J or F by one, and in order to reduce the problem to manageable proportions we must make some simplifications and approximations, always bearing in mind the accuracy of the experimental observations. The remainder of this section is concerned with how this aim is achieved in practice. Excepting molecules that possess stereoisomers, the rotational wave functions of asymmetric top molecules have a well defined parity which is retained in the presence of an external magnetic field. The parity of a function is related to its behavior under the space-fixed inversion operation E* E*F(X,

Y,Z)=F(-X,

-Y,

-Z),

(43)

where X, Y, 2 are the space-fixed coordinates. The function may be either unchanged or change and is referred to as having + or - parity, respectively. For sign under the E* operation molecules belonging to symmetry groups containing the E * operation (planar asymmetric tops), the parity of the wavefunction is easily defined; the effect of the E * operation on the rotational basis functions (20) is [38] E*ITJNK~) and eigenfunctions ]$vKm

=

(-)“-“IqN-

of E* with eigenvalues

&-) = (2)-l”{

(44

Km) + 1 are

]TjNKm) *( -)N-Kl$v

- Km)].

(45)

Asymmetric top levels specified by NKoK, are combinations of these basis functions and have parities determined by (- 1) Kc for planar molecules. Oka [39] treats the case of molecules belonging to symmetry groups containing permutation-inverstion operations but not E *. In this case, the parity label and the overall symmetry label of the wavefunction [40] are equivalent. Bunker has pointed out [40] that while this is true, the overall symmetry label is often more useful because it gives additional information necessary for the calculation of nuclear statistical weights for example. A knowledge of the parity of a molecular eigenstate is important because the effective Hamiltonian (1) connects only states of the same parity whereas electric dipole transitions occur between levels of opposite parity. Use of eq. (45) makes it straightforward to set up the Hamiltonian matrix in a parity conserving basis set. There are no non-zero matrix elements between functions of opposite parity and hence the size of the Hamiltonian matrix is reduced by a factor of two. There are several advantages to using a parity conserving basis set over the more usual modified Wang combinations [9]. In the latter scheme, matrix elements of the Hamiltonian off diagonal in N mix states of opposite symmetry whereas this is not the case for the true parity labels. Secondly, in a simple k combination basis set, the levels connected by electric dipole transitions may be of the same or different sign depending on the quantum number changes. The program described here and in ref. [51] uses the true parity conserving functions as its basis. The division of the Hamiltonian using basis functions of a well defined parity is a rigorous one; there is no loss of accuracy in the model resulting from this transformation. When the

T.J. Sears / Energy levels of an asymmetric top free radical

19

molecule possesses orthorhombic symmetry we may further factorize the Hamiltonian matrix since there are no terms in the operator which mix vectors differing in K by an odd number. This means that the matrix may be blocked into two parts one containing the even K and the other the odd K basis vectors. For many applications, however, we may further reduce the size of the matrices with which we have to deal by making approximations that do not significantly affect the eigenvalues. Firstly, although the separation of the matrix into odd and even K blocks is only rigorously good for asymmetric top molecules belonging to higher s~et~ groups, the effects of matrix elements connecting basis states of odd and even K in other cases are often negligible compared with the experimental precision. The matrix elements in question arise from terms in the effective spin-rotation, dipolar hyperfine and Zeeman operators. The only situation where they affect the calculated energy levels in practice is where there is an accidental near degeneracy between two interacting levels. The best known examples occur in near prolate asymmetric top molecules where close coincidences can occur between a rotational level IN, K, J) and the levels ]N & 1, K + 1, J). This is the case for the unsymmetrical triatomics HO, [lS], DO, [41] and HCO [25] where the (small) shifts in the energy levels have been used to determine some of the parameters describing these effects. For most applications, however, the Hamiltonian matrix may be divided into odd and even K blocks with no significant loss of accuracy. By making use of the symmetry arguments above we see that we may reduce the size of the matrix required to calculate the molecular eigenvalues and eigenfunctions by a factor.of about four in most cases. Strictly speaking, when we include the molecular Zeeman interaction we will still have to deal with an infinite sized matrix because, in this case, the only good quantum number is MF and all states with total angular momentum, F, equal to or greater than I.M,I will contribute. Fortunately, we can always bring the size of the Hamiltonian matrix down to manageable proportions by truncating the basis set at a point where the inclusion of further basis states causes changes to the eigenvalues which are smaller than the experimental uncertainty. For simplicity, we first consider the arguments for the case where there is no external magnetic field. Here, the quantum number, F, representing the total angular momentum, is good. However, the size of the matrix increases rapidly with F, that is as we go to higher rotational levels, because we have to include all basis states of different N, K and J which give rise to a particular hyperfine level with quantum number F. Some reduction in the size of K-basis can usually be made if the molecule approximates more or less closely to a symmetric top. In this case, the effect of matrix elements connecting basis states differing in K are small and K approximates to a good quantum number. Under these circumstances we can cut off the K basis at K’ - K = AK = 0, rt 2, or +4,... depending on how asymmetric the molecule is and how accurately we require to calculate the eigenvalues. The effects of K-basis truncation are illustrated in table 1 in a calculation of the energies of some rotational levels of NH, in its ground %‘B, state. The structure of this molecule is such that it is a strongly asymmetric rotor (asymmetry parameter K= - 0.384) and the energy levels are expected to exhibit marked basis set truncation effects. This is indeed the case as evidenced by the entries in table 1. These are the calculated energies of various levels with rotational quantum number N = 6. To avoid repetition, the calculations performed neglect the hyperfine splittings, due to the 14N (I = 1) and ‘H( I = l/2) nuclei, however, the fine structure splittings, arising from the spin-rotation interaction in this case, are included. All calculations included basis states with N’ = N, N rfr1 that is N’ = 5, 6 or 7 here, since these are mixed in first order by

T.J. Sears / Energy levels of an asymmetric top free radical

20

the spin-rotation Hamiltonian operator. The columns of table 1 are the calculated energy levels for various different truncations of the K-basis. For NH, in these levels, it is clear that cutting off the basis set at /AK\ < 4 in the calculation yields a very good approximation to the eigenvalue for all but the 6,, level for which an increased, IAKl K 6, basis set is required. Close examination of the calculated levels reveals that although the molecule is nominally closer to the prolate top limit (K = - l), in the levels considered here the energy levels are better labelled by their K, rather than K, value corresponding to the oblate top (K = + 1) limit. This effect arises due to the combined rotational and centrifugal distortion mixing of the symmetric top basis functions and an examination of the eigenvector coefficient matrix reveals that the eigenstates have large contributions from many different prolate symmetric top functions used as the basis vectors. The quantum number K, becomes less good with increasing rotational excitation and for these and higher levels in such molecules a better approach would be to adopt a basis set of oblate symmetric top functions, since the size of the basis set required for the reliable calculation of the energy of any given level will be smaller. Apart from the considerations of efficiency discussed above, there is another problem encountered in the calculation of higher lying rotational energy levels in NH, and similar strongly asymmetric rotors. This involves the labelling of the energy levels. The program relies upon recognizing the required eigenvalue corresponding to a specified level through an examination of the eigenvector coefficient matrix. In some cases, there is often such extensive mixing of the basis vectors that this procedure breaks down and either an incorrect eigenvalue is picked up or none is selected. In NH, in its ground state, such scrambling occurs at around N = 8 for levels

Table 1 Effects of basis set truncation Rotational level

6, S F2

61, F, F2

61, FI F2

62s Ft F2

62, Ft F2

b,

on various N = 6 rotational Calculated

levels of A2B, NH, a)

energy with basis including

IAKl&2

IAKl<4

lAK1$6”’

394.98876 395.09757

393.07251 393.18309

393.06694 393.17755

393.56222 393.67549

393.39278 393.50642

393.39278 393.50642

479.47776 479.74995

478.69323 478.96799

478.69323 478.96799

485.34227 485.66692

485.29490 485.61971

485.29490 485.61971

532.70001 532.06296

532.63278 532.99607

532.63278 532.99607

a) The entries in the table are calculated rotational energies of the levels in cm -r for different basis set sizes. All calculations included basis sets with IANI Q 1 and the molecular parameters used were those quoted by Kawaguchi et al. [38]. Hyperfine interactions due to the 14N and ‘H nuclei were ignored in these calculations. b, Fr and F, label the fine structure levels. F, corresponds to a level with J = N + f and F2 to one with J = N - f . ‘) For these levels, a basis including vectors with jAK1 Q 6 is complete.

T.J. Sears / Energy levels of an asymmetric top free radical

21

with small K, and use of the program under these circumstances requires care. Fortunately, these effects are confined to light, strongly asymmetric, rotors. In practice this means triatomic dihydrides, and in all other cases tested to date, no problems have been encountered. Similar arguments can be used to decide on the appropriate basis set cut off points for a molecule subject to an external magnetic field. The terms in the Zeeman Hamiltonian are generally smaller than the rotational and fine structure contributions in the zero-field Hamiltonian and a basis that includes all the various fine and hyperfine states for the N and K states included in the zero field basis will normally provide acceptable accuracy. The program described in detail in the next two sections follows the procedures outlined above in order to reduce the size of the Hamiltonian matrix as far as possible. The size of the basis set used is defined by the user in terms of the number of N and K states to be included. In order to reduce the amount of computation involved in setting up the matrix, the matrix elements of the rotational, centrifugal distortion and fine structure Hamiltonians have been explicitly calculated from the general expressions given in eqs. (27)-(39) and coded algebraically. This procedure avoids the need for computation of large numbers of 3, 6 and 9-j vector coupling coefficients and the matrix elements are sufficiently few in number and simple in form [9,11] that coding and checking each individually is possible.

4. Machine implementation We now consider in more detail how the molecular energy levels as represented by the eigenvalues of the effective Hamiltonian are calculated. In this section we discuss the problems associated with setting up the basis set for the calculation of the energy of a particular level, computation of transition frequencies and relative intensities and, for magnetic resonance transitions, the calculation of the tuning rate of the transition in the field. This last quantity gives information on the expected spectral linewidths and is a useful tool in spectrum analysis. Optimization of the molecular parameters to fit to observed transition frequencies requires calculation of the derivative of the transition frequency with respect to the parameter to be varied and we also discuss how this is achieved in practice. The first problem involved in calculating a particular energy level is to set up the appropriate basis set. The quantum numbers which label the level in question are the rotational angular momentum N, its projection on the symmetric top axis K, the parity of the level, the fine structure quantum number J, or the total angular momentum quantum number F, and the number identifying the particular hyperfine state (see below). We also need to specify the basis set limits in terms of the maximum difference in N and K values between any basis state and the level in question. Taking into consideration the multiplicity of the state, i.e. the value of S, and the nuclear spin I, the program calculates the quantum numbers for the basis vectors given the vector coupling rules J=(N-Sl,

IN-S+lI,...,

N+S,

F=IJ-I(,IJ-I+lI,...,

J+I.

If the calculation is to include the effect of an external magnetic field, all the possible states (F-values) for a given MF and range of N, K and J values are set up. Otherwise,

(46) hyperfine since F is

T.J. Sears / Energy levels of an asymmetric top free radical

22

Table 2 Basis vector quantum numbers for various N = 9 energy levels of CH, X’B,

a)

(i) 909fine structure levels b, J=8

J=lO

J=9

N

K

I

N

K

J

N

0

8

2 4 2 4 0 2 4

8 8 8 8 8 8 8

8 8 9 9 9 10 10

2 4 0 2 4 2 4

9 9 9 9 9 9 9

9 9 9 10 10 11 11 11

K

J 10

10 10 10 10 10 10 10

(ii) 9,s hyperfine structure levels ‘) F=7

F=8

F=9

F=lO

F=ll

N

K

J

N

K

J

N

K

J

N

K

J

N

K

J

7 7 7 7 7 7 7 7 7 7 8 8 8 8 8 8 9 9 9

I 3 5 5 1 3 5 1 3 5 1 3 5 1 3 5 1 3 5

6 6 6 6 7 7 7 8 8 8 7 7 7 8 8 8 8 8 8

7 7 7 7 7 7 7 8 8 8 8 8 8 8 8 8 9 9 9 9 9 9 10 10 10

1 3 5 5 1 3 5 1 3 5 1 3 5 1 3 5 1 3 5 1 3 5 1 3 5

7 7 7 7 8 8 8 7 7 7 8 8 8 9 9 9 9 8 8 9 9 9 9 9 9

7 7 7 7 8 8 8 8 8 8 9 9 9 9 9 9 9 9 9 10 10 10 10 10 10 11 11 11

1 3 5 5 1 3 5 1 3 5 1 3 5 1 3 5 1 3 5 1 1 5 1 3 5 1 3 5

8 8 8 8 8 8 8 9 9 9 8 8 8 9 9 9 10 10 10 9 9 9 10 10 10 10 10 10

8 8 8 8 9 9 9 9 9 9 10 10 10 10 10 10 10 10 10 11 11 11 11 11 11

1 3 5 5 1 3 5 1 3 5 1 3 5 1 3 5 1 3 5 1 3 5 1 3 5

9 9 9 9 9 9 9 10 10 10 9 9 9 10 10 10 11 11 11 10 10 10 11 11 11

9 9 9 9 10 10 10 10 10 10 11 11 11 11 11 11 11 11 11

1 3 5 5 1 3 5 1 3 5 1 3 5 1 3 5 1 3 5

10 10 10 10 10 10 10 11 11 11 10 10 10 11 11 11 12 12 12

a) Basis includes states up to and including AN = +2 and AK = +4. b, 9as is associated with para-CH, which has a net proton nuclear spin of 0. ‘) 9,, is associated with ortho-CH, which has a net proton nuclear spin of 1.

T.J. Sears / Energy levels of an asymmetric top free radical

23

the good quantum number, only the basis states with the given F value are generated. The basis vectors are stored in a real (n x 4) array with the four columns containing the N, K, J and F values, respectively. Each row of the array therefore corresponds to a basis vector of the form given in eq. (26). Examples of typical basis sets are given in table 2. This basis vector array is then used to construct the Hamiltonian matrix using the elements given in the last section together with the identity (45) to convert to a parity conserving basis. Each element Hii of the Hamiltonian matrix is the matrix element of the operator (1) between basis state i and basis statej. For physical problems, the Hamiltonian matrix is Hermitian so that Fij = HjT and all matrix elements are real. The matrix representations of the zero-field Hamiltoman and the Zeeman Hamiltonian are constructed separately then added together prior to diagonalization if the calculation requires the inclusion of the Zeeman effect. Diagonalization is accomplished by a library routine that uses a Householder transformation to bring the matrix to tridiagonal form followed by the QL algorithm [42] which completes the process. The eigenvalues come out in ascending order in a one dimensional array, E, and elements Ei while the eigenvectors are given as the eigenvector coefficients in a two dimensional array S such that Sji (j = 1, n) contains the coefficient of basis vector j in the eigenstate i corresponding to the eigenvalue E, . A problem which is often encountered is in deciding which eigenvalue corresponds to the level of interest since the order of eigenvalues in E bears no relation to the order of the vectors in the basis set. When the molecule in question has many fine and hyperfine levels, deciding on unambiguous quantum number labels during a calculation is often quite difficult. This is especially true when the zero-field states are extensively mixed by matrix elements of the Zeeman Hamiltonian in a magnetic field. One could, of course, resort to a slower diagonalization method and retain the ordering of the basis vectors but this becomes increasingly expensive in terms of machine time for larger matrices, which is precisely where the identifications are the most difficult. The method adopted in the present application is to use the eigenvector coefficient matrix to pick out eigenstates with the correct N and K and simply number the fine and hyperfine structure states counting from the lowest energy one. A particular eigenstate is then labelled by its N, K, parity and F (or MF when the external magnetic field is not zero) and a hyperfine state label which is an integer number. As discussed in the previous section, the use of the eigenvector coefficient to identify the N and K quantum numbers of the eigenstates is reliable in all but the most difficult cases. Calculation of a molecular transition frequency thus proceeds by repeating the steps outlined above for the two states connected in the transition and subtracting the lower eigenvalue from the upper. If the transition connects different vibronic states, that is, it is not a pure rotational one, then each eigenvalue calculation uses a different effective Hamiltonian. The intensity of an electric dipole transition between two states is proportional to the matrix element of the dipole moment operator and the radiation field strength. Intensities of vibration-rotation transitions depend on the derivative of the dipole moment function with respect to the vibrational coordinate but as far as the rotational dependence is concerned, the method is the same. In irreducible tensor notation, the relative intensity of a transition between state la) and state Jb) is given by c l(~lTd(p)*T’-,(E)lb)l*. P

(47)

24

T.J. Sears / Energy levels of an asymmetric top free radical

The eigenstates la) and lb) are linear combinations of the basis functions: Iu) = c ci,~7@iKjs;Jlri~~~))

J

Making this substitution relative intensity as: IaC

x

and evaluating the matrix element we derive an expression for the

T’,(E)CCi,C,,(-)~-M’l P

(48)

lb) = c cj,~~iNiKisjJirifjM~).

i

ij

[

[ (21;1+

i

1)(2F, + 1)]1’2

X C T,‘(P)(

-)N,-K’[(2N,

1 _p

_LF

{4 +

:

1)(2Nj

I;; MF (-)‘+I+‘+l / I

I

(-1 w+s+J,+i[(2JI+

:

+

‘)I

‘q

_z;

i

1)(2.5-t

1)]1’2

J”

i,

:I

I gl’.

J

’

4

(49) Here, T:(r) are the irreducible tensor components of the dipole moment operator in the molecule fixed axis system. Absolute intensities may be calculated by including a factor depending on the population difference between the states. Eq. (49) refers to the intensity of a single MF, + M4 component and where individual components are not resolved, for example in the case of a zero-field experiment, the intensity is given by the sum of the individual component transitions. Finally, it should be noted that the operator Hs = T’(p) T’(E) which represents the Stark interaction between a molecule with dipole moment T’(p) and an electric field T’(E) is identical to that evaluated above and the Stark interaction energy can be calculated by setting up the matrix of Hs using the matrix element expression given in eq. (49). In the assignment of magnetic resonance spectra, it is often useful to be able to calculate the rate at which the transition frequency, V, is tuning in the magnetic field, B,, to enable comparison with experiment. We can simply relate this quantity to the rate at which the two energy levels tune since l

av -=

aE, -- aE,

aBo

aB,,

aB,'

where E, and Eb are the energies of the levels involved in the transition. The problem is resolved into the calculation of the partial derivatives of the energy levels with respect to the magnetic flux density. These quantities are very similar to those required in the least squares optimization of the parameters in the effective Hamiltonian where we require av/i!lq,, where pj is the jth parameter for example. This type of derivative is efficiently calculated using a method described by Castellano and Bothner-by [43] who related the derivative of the ith eigenvalue of the Hamiltonian matrix with respect to the jth parameter to the same derivative in the basis representation by aEi

-@z&p

aPj [

‘ap, l

1

;,

(51)

T.J. Sears / Energy levels of an asymmetric top free radical

25

Here S is the eigenvector coefficient matrix whose elements are defined in eq. (48) and (aH/ap,) is the derivative of the Hamiltonian matrix in the basis representation with respect to the jth parameter. Generation of the derivative matrix in the basis representation is straightforward and the derivative of the ith eigenstate according to eq. (51) is the ith diagonal element of the matrix product above. Returning to the problem of calculating the tuning rate of a magnetic resonance transition, the derivative of the Hamiltonian matrix with respect to the magnetic flux density in the basis representation is simply the matrix of the Zeeman operator with the flux density, B,,, set to unity, and a relationship analogous to eq. (51) can be used.

5. Description of package In this section we move from a general consideration of the theory of the calculation to the specific details of how it is implemented in fact. We begin by listing and explaining briefly the different types of calculation which are possible, then proceed to describe them in detail. Examples of some calculations are included. Finally, we summarize the uses of this type of program and offer some concluding remarks. Further details of the structure of the program and a complete listing are available in a companion article [51]. There are four primary modes of operation of the program although several alternative options exist in some of them. The first, and simplest, mode calculates eigenvalues for a chosen set of rotational levels using given molecular parameters for one or two states. The second type of calculation involves calculating frequencies, optionally with intensities, for given molecular transitions. For magnetic resonance transitions, this type of calculation will also estimate a magnetic flux density at which the resonance will occur given an estimated operating frequency. Related to this kind of calculation is the third type which is only relevant for magnetic resonance data. All resonances for a particular molecular rotational (rovibrational) transition between given magnetic field limits are identified for a series of operating frequencies. This type of calculation is useful in assigning gas phase magnetic resonance spectra and is especially powerful when combined with the graphical techniques [21] which have proved to be the most effective in assigning complex laser magnetic resonance spectra. Finally, having assigned the spectra, we may choose to refine the Hamiltonian parameters by a least squares fit to the observed data. The program developed here allows simultaneous fitting of vibration-rotation and pure rotational spectra, for appropriate weighting of each data point, and for mixing both zero-field data, that is taken in a swept frequency experiment such as conventional microwave spectroscopy, and magnetic resonance data. We first consider the simplest type of calculation, computation of eigenvalues given a set of molecular parameters. Apart from the parameters themselves, we require to know the multiplicity of the electronic state of the molecule and the nuclear spin of the nucleus with non-zero spin. In its present form the program allows for only one nucleus with non-zero spin. The basis set description is also required, in terms of the maximum allowable differences in N and K in the basis states used in the calculation of any particular eigenstate. The range of rotational quantum numbers for which eigenvalues are required is also needed as is an indication as to whether matrix elements with AK = + 1 have to be included. With this information, the program sets up the basis set and matrix for each fine and hyperfine structure state and computes the eigenvalues.

26

T.J. Sears

/ Energy

levels of an asymmetric

top free radical

If a variable representing the T, value for the vibration-rotation transition is non-zero, the calculation will proceed for a second state, represented by a second set of molecular parameters, which is by default assumed to be the lower lying state. The second mode of calculation, which was originally written to test trial assignments of magnetic resonance data, was later developed to include calculations for pure rotational and vibration-rotational transitions occurring in the absence of any external magnetic field. For magnetic resonance transitions, calculations such as those discussed immediately below have proved more useful. In the current mode, two complete sets of molecular parameters are required, the first representing the upper state, the second representing the lower state involved in the transitions. For pure rotational transitions, the two are identical. Also required are the usual descriptions of the basis set to be used and the molecule’s net electronic and nuclear spin. The program then runs through a set of data giving the detailed quantum number changes for the transitions, including the fine and hyperfine quantum number assignments, an estimated frequency for the transition and, for magnetic resonance transitions, the estimated magnetic flux density. The estimated frequency is strictly only required in the case of magnetic resonance transitions where it is used to estimate a value for the magnetic flux density at which the resonance will occur. The input magnetic flux density is used by the program for the eigenvalue calculation and for estimating the transition tuning rate as described in the previous section. An estimated frequency at which the transition would occur for the chosen magnetic flux density is then generated. Finally, the relative intensity of the transition may also be requested and will be calculated assuming a Boltzmann distribution amongst the molecular energy levels appropriate to an input temperature. The third mode of calculation applies only to magnetic resonance type transitions. For a molecular transition specified only by the rotational (N, K and parity) quantum numbers of the states involved, all resonances which occur at a given operating frequency between specified field limits are calculated for all possible fine structure transitions between the rotational levels in question. Hyperfine structure is ignored in this calculation since it is usually straightforward to recognize hyperfine splittings in observed spectra. The calculation proceeds in the following way. For each of the levels, a matrix containing the eigenvalues for all possible MJ values at a series of values of the flux density between the limits specified by the input data is generated. A search through these matrices is then performed to look for allowed transitions occurring close to the input operating frequency. When a candidate is found, the eigenvalues of the levels involved are fit to a quadratic function in the magnetic field and an interpolation used to compute the estimated resonant field. When all resonances for a particular operating frequency have been identified the calculation continues by looking for a new operating frequency, new magnetic field limits or new rotational quantum numbers. This type of calculation has proved to be very useful in the assignment of laser magnetic resonance data for several species including CH, [21] and HSO [44] recently. When combined with computed relative intensities and graphical presentation of the results [22] it becomes a very powerful assignment tool. Finally, the parameters in the effective Hamiltonian may be refined to fit to observed data. Parameters are input for two different states followed by blocks of data referring to infrared, i.e., vibration-rotational, and pure rotational transitions. The parameters that are to be varied are identified in a single dimensional integer array whose elements contain the value one or zero depending on whether a particular parameter is to be floated or kept fixed in the least squares

T.J. Sears / Energy levels of an asymmetric top free radical

21

procedure. Each observation is accorded a weight which is normally set at the square of the inverse of the experimental uncertainty and each observation is associated with its own basis set, so that the size of the basis set used for each data point can be defined according to its estimated precision. The program fits to either or both magnetic resonance and zero-field spectroscopic data and in both cases, fits to the frequencies of the observations. For a magnetic resonance experiment at a particular frequency it might be preferable to fit to fields since these are the experimentally observed quantities. In this case the experimental linewidths are proportional to the inverse of the tuning rate of the transition, (av/aB)-‘. Hence the relative uncertainty of measurement is also proportional to (av/aB)-’ and the appropriate weight of a data point in a least squares fit is @v/U)*. In a fit to frequency, the uncertainty of measurement in frequency terms is also related to the tuning rate @v/U?) and the same result can be obtained if the least squares weight of each data point is equal. Optimization of the molecular parameters proceeds by the usual methods [45]. Although the molecular frequencies do not depend linearly on the parameters, we make the assumption that for small changes in the parameters, proportional changes occur in the frequencies and therefore determine a series of best fit changes to the initial parameters that minimize the differences between the observed and calculated frequencies in a least squares sense. The procedure is then repeated and normally converges within three cycles. The matrix inversion required in the least squares procedure is performed by a suitable local library routine. 6. Analysis of the millimeter and submillimeter wave spectrum of the HCO radical The formyl radical, HCO, was the first unstable non-linear radical detected by gas phase magnetic resonance spectroscopy [13] and its importance in the chemistry of hydrocarbon flames has encouraged a considerable number of further high resolution studies [20,25,46-501. The most complete analysis has been performed by Blake and coworkers [20] who measured part of the millimeter and submillimeter wave spectrum of the radical. Independently, Brown, Radford and Sears measured and analyzed an extensive set of FIR LMR data [25] which contain complementary information on the rotational energy level structure in the ground state of the radical. In order to extract the best set of molecular parameters describing this state, it is necessary to combine the data from these sources together with the earlier microwave and EPR measurements [50] and the FIR LMR measurement reported by Cook et al. [49]. The FIR LMR dataset is very extensive and in order to reduce the amount of computation required, the parameters of Brown, Radford and Sears [25] were used to make best estimates of the zero-field transition frequencies, including hyperfine splittings, for the transitions observed in the magnetic resonance experiments. These reduced frequencies, which are estimated to be reliable to 2-3 MHz, are given in table 3. The data in table 3 together with the previously reported measurements [20,50] were fit to the Hamiltonian described in section 3. The data were accorded weights according to the inverse square of the estimated precision of the measurement. Although HCO is a light molecule, it is fairly close to the prolate symmetric top limit and the maximum basis sets used in the calculations were IANI < 1 and IAKl G 4 1’mrits which were empirically found to be quite adequate to calculate the levels to within their experimental precision. The results of the fit are summarized in tables 3 and 4. As can be seen from the residuals in table 3, the data are nearly all

T.J. Sears / Energy levels of an asymmetric top free radical

28 Table

3

Estimated Rotational *

K,

8

5

4

11 11 11 11 11

2

2

1

zero-field

parity

+

_

+

+ + + + +

1

-

9 10

8 6

intervals in %*A’ HCO a)

transition b,

11

9

rotational

+ + -

+ -

+

+

F

i

+

N

K,

parity

1

_

F

i

Frequency

Frequency

(cm-‘)

(obs-calc)X103

7

1

6

1

92.63199

-0.20

8

1

7

1

92.63480

- 0.16

8

2

7

2

92.76270

0.15

9

1

8

1

92.76569

0.19

4

1

5

1

51.71828

- 0.10

7

6

1

+

5

1

6

1

51.71985

- 0.09

5

2

6

2

51.94146

0.09

6

1

7

1

51.94325

0.12

3

1

2

1

33.90929

- 0.02 - 0.06

3

0

-

4

1

3

2

33.90571

4

2

3

1

33.98343

0.02

5

1

4

1

33.97971

- 0.03

10

1

10

2

31.69731

- 0.03

11

1

11

1

31.68436

- 0.09

10

1

10

2

31.55448

-0.38

11

1

11

1

31.54156

- 0.45

10

1

10

2

31.55381

- 0.37

11

1

11

1

31.54089

- 0.43

10

1

10

2

31.33408

0.25 =)

11

1

11

1

31.32115

0.20 =)

10

1

12

1

10

1

12

1

1

2

2

1

8

1

9

1

8

1

9

1

9

1

10 10 10 10 10

2 3 3 4 5

_ + _ -

10

6

_

0

0

+

8 8 10

3 4 0

_ +

9

1

31.82675

11

1

31.75276

- 0.35 c’ 0.37 =)

9

1

31.83023

- 1.07 c, 0.97 c,

11

1

31.72866

0

1

25.79157

0.02

1

1

25.78704

- 0.01

8

2

25.70415

- 0.40 c,

9

1

25.69121

- 0.46 =)

8

2

24.43789

0.78 c,

9

1

25.42492

9

1

25.53048

- 0.08

0.72 =)

10

1

10

2

25.52735

- 0.07

10

2

10

1

25.58992

0.60

11

1

11

1

25.58696

0.62

8

2

7

_

7

1

23.13786

0.34

5

1

6

+

5

1

23.82402

- 0.08

6

1

6

2

23.82122

-0.11

6

2

6

1

23.89563

7

1

7

1

23.89300

0.12

0

1

0

1

22.70329

- 0.01 - 0.06

-

0.14

1

1

1

2

22.69942

1

2

1

1

22.99564

0.01

2

1

2

1

22.99247

- 0.01

2

1

2

1

23.07241

- 0.04

3

1

3

2

23.06916

- 0.08

3

2

3

1

23.18856

0.04

4

1

4

1

23.18550

0.00

_

29

T.J. Sears / Energy levels of an asymmetric top free radical Table 3 (continued) Rotational N

K,

transition parity

8

2

-

8

2

+

8

3

+

8

3

-

8

4

+

2

1

-

Frequency (cm-‘)

b, F 7 8 7 8 7 8 8 9 7 8 8 9 7 9 1 2 2 3

i

+ 1 1 1 1 1 1 2 1 1 1 2 1 1 1 1 1 2 1

N

parity

K, 7

2

+

7

2

-

7

3

_

7

3

+

7

4

-

2

0

+

F

i 7 8 7 8 6 7 7 8 6 7 7 8 6 8 1 2 2 3

2 1 2 1 1 1 2 1 1 1 2 1 1 1 1 2 1 1

22.97996 22.96700 23.00340 22.99043 23.16431 23.16426 23.10976 23.10967 23.16445 23.16439 23.10990 23.10983 23.18210 23.08813 22.89922 22.89570 23.06028 23.05702

Frequency (obs-calc)X103 -0.17 - 0.22 - 0.27 - 0.33 - 0.03 - 0.02 0.09 0.07 - 0.03 - 0.03 0.08 0.09 - 0.21 Q 0.20 =) - 0.03 - 0.08 0.02 - 0.01

a) From data given in ref. [25]. b, Rotational levels are labelled by quantum numbers N, K,, parity, F and a number i representing the number rotational level of the given F for a particular N, K, and parity counting from the lowest energy one. ‘) K-doubling not resolved in this transition.

Table 4 Molecular

parameters

for A2A’ HCO (in cm-‘)

A = 24.329036(11) a)

co.

A, = 0.0306891(96) 104A vK = 0.1520(13) 105A, = 0.3952(19) 1036, = 0.1499(16) 1066, = 0.3911(43) lo%,, = -0.4718(87)

Cbb + Ecc = 0.381448(40) - ebb - c,, = 0.781705(67)

+

2r,,

+(B + C) =1.4463102(4) +( B - C) = 0.047648(32)

‘bb

-

c~~

=

0.0074878(47)

0.01343(15) hb ASK= - 0.001637(12) 105(As,, + ASKN)= -0.332(73) 106A=,,, = 0.210(33) lo5 ‘D; = O/445(88) +

Ebaj =

b, a = 0.0129667(83) 103aa, = 0.3746(96) 103(bb, - ccI) = 0.580(18)

*) Numbers in parentheses are one standard deviation of the least squares fit. b, All other sextic and higher centrifugal distortion constants constrained to zero.

of the

30

T.J. Sears / Energy levels of an asymmetric top free radical

fit to within experimental precision. The largest residuals are associated with the K = 6 levels in the LMR data where the (observed - calculated) frequency is of the order of 30 MHz. Additionally, it appears that the errors are systematic in the spin-rotation splitting. The introduction of the higher order spin-rotation coupling constant @SChelped but did not completely eradicate the problem. The data points in question derive from relatively few observed resonances but it is felt that their assignments are certain. It is possible that for these high K rotational levels, the model Hamiltonian is not adequate due to the neglect of terms derived from a Renner-Teller interaction between the ground and low lying A’A” electronic state. The first noticeable effects of such an interaction would be in these rotational levels. A definitive answer will have to await additional measurements of the spin-rotation splittings in these high lying rotational levels. No such problems were encountered in the millimeter wave data which include information on relatively low lying rotational levels. All of the data were fit to their expected precision. The derived parameters are given in table 4 and all the major constants are well determined. The effects of matrix elements of the spin-rotation and dipolar hyperfine separations which connect basis states differing in K by one are quite important for some levels, however, it did not prove possible to determine separately the off diagonal components of the relevant two tensors, presumably due to a lack of sufficient data. Arbitrarily, Tabwas constrained to zero in the final analysis. The resulting parameters are certainly the best available for the ground electronic state of HCO and should reliably reproduce the rotational energy level structure over a wide range of rotational quantum numbers.

7. Summary In this review, I have outlined the form of the model used in fitting high resolution spectral data for asymmetric top molecules. Although such data can always be fit to a power series expansion in the angular momenta, the effective Hamiltonian model has the advantage that the parameters determined can be readily interpreted in terms of the details of the internal molecular structure since they are explicitly related to admixtures of other vibronic states. Various structural theories can therefore be tested by trial calculations of the effective parameters determined in a fit to experimental data. The computer program, described in ref. [51], is the result of several years of periodic development and is very thoroughly tried and tested. I have tried to anticipate problems where possible and likely problem areas are discussed in the text together with possible solutions. The overall flexibility of the package means that it is applicable to a wide cross section of gas phase spectroscopic data and current interest in the spectroscopy and structure of radicals and ions suggests that it will retain its usefulness and applicability.

Acknowledgements I am extremely grateful to Dr. John M. Brown for his advice, encouragement and many helpful suggestions during the time that this work was performed. I am also indebted to Dr. Philip R. Bunker for inviting me to write this article and for his constructive comments on the manuscript which helped to clarify several points. Research carried out in part at Brookhaven National

T.J. Sears / EnergV levels of an asymmetric top free radical

laboratory under DE-AC0276CH00016 with the US Department Division of Chemical Sciences, Office of Basic Energy Sciences.

of Energy and supported

31

by the

References [l] [2] [3] [4] [5] [6] [7] [8] [9] [lo] [ll] [12] [13] [14] [15] [16] [17] [18] [19] [20] [21] (221 (231 [24] [25] [26] [27] [28] [29] [30] [31] [32] [33] [34] [35] [36] [37] [38] [39] [40] [41]

For a review see A. Carrington, D.H. Levy and T.A. Miller, Advan. Chem. Phys. XVIII (1970) 149. H.E. Radford, K.M. Evenson and C.J. Howard, J. Chem. Phys. 60 (1974) 3178. J.T. Hougen, H.E. Radford, K.M. Evenson and C.J. Howard, J. Mol. Spectrosc. 56 (1975) 210. C.E. Barnes, J.M. Brown, A. Carrington, J. Pinkstone, T.J. Sears and P.J. Thistlethwaite, J. Mol. Spectrosc. 72 (1978) 86. K.M. Evenson, Faraday Disc. Roy. Sot. Chem. 71 (1981) 7. A.R.W. McKellar, Faraday Disc. Roy. Sot. Chem. 71 (1981) 63. J.H. van Vleck, Rev. Mod. Phys. 23 (1951) 213. R.F. Curl, Jr., Mol. Phys. 9 (1965) 585. W.T. Raynes, J. Chem. Phys. 41 (1964) 3020. T.A. Miller, Mol. Phys. 16 (1969) 105. J.K.G. Watson, in: Vibrational Spectra and Structure, vol. 6, ed. J.R. Durig (Elsevier, New York, 1977) p. 1. J.M. Brown and T.J. Sears, Mol. Phys. 34 (1977) 1595. I.C. Bowater, J.M. Brown and A. Carrington, Proc. Roy. Sot. Lond. A333 (1973) 265. D.M. Brink and G.R. Satchler, Angular Momentum (Oxford Univ. Press, London, 1962). A.R. Edmonds, Angular Momentum in Quantum Mechanics (Princeton Univ. Press, Princeton, 1960). S. Kirschner, Ph.D. Thesis, Ohio State University (1978). R.N. Dixon and G. Duxbury, Chem. Phys. Lett. 1 (1967) 330. J.M. Brown and T.J. Sears, J. Mol. Spectrosc. 75 (1979) 111. K.-E. J. Hallin, Y. Humada and A.J. Merer, Can. J. Phys. 54 (1976) 2118. G.A. Blake, K.V.L.N. Sastry and F.C. DeLucia, J. Chem. Phys. 80 (1984) 95. T.J. Sears, P.R. Bunker, A.R.W. McKellar, K.M. Evenson, D.A. Jennings and J.M. Brown, J. Chem. Phys. 77 (1982) 5348. R.F. Curl, Jr. and J.L. Kinsey, J. Chem. Phys. 35 (1961) 1758. G.W. Hills, J.M. Cook, R.F. Curl and F.K. Tittel, J. Chem. Phys. 65 (1976) 823. W.H. Flygare and R.C. Benson, Mol. Phys. 20 (1971) 225. J.M. Brown, H.E. Radford and T.J. Sears, to be published. J.C.D. Brand and C.G. Stevens, J. Chem. Phys. 58 (1973) 3331. F.W. Birss, D.A. Ramsay and S.M. Till, Chem. Phys. Lett. 53 (1978) 14. K.H. Fung and D.A. Ramsay, J. Chem. Phys. to be published. A.R.W. McKellar, P.R. Bunker, T.J. Sears, K.M. Evenson, R.J. Saykally and S.R. Langhoff, J. Chem. Phys. 79 (1983) 5251. T.J. Sears, P.R. Bunker and A.R.W. McKellar, J. Chem. Phys 75 (1981) 4731. T.J. Sears, P.R. Bunker and A.R.W. McKellar, J. Chem. Phys 77 (1982) 5663. P. Jensen, P.R. Bunker and A. Hoy, J. Chem. Phys. 77 (1982) 5370. J.T. Hougen, P.R. Bunker and J.W.C. Johns, J. Mol. Spectrosc. 34 (1970) 136. P. Jensen, Comput. Phys. Rep. 1 (1983) 1. A.R.W. McKellar, J. Chem. Phys. 77 (1979) 81. R.S. Lowe and A.R.W. McKellar, J. Chem. Phys. 74 (1981) 2686. G. Herzberg, Molecular Spectra and Molecular Structure, vol. I (Van Nostrand-Reinhold, New York, 1950). P.R. Bunker, Molecular Symmetry and Spectroscopy (Academic Press, New York, 1979). T. Oka, J. Mol. Spectrosc. 48 (1973) 503. P.R. Bunker, in: Vibrational Spectra and Structure, vol. 3, ed. J.R. Durig (Dekker, New York, 1975) chap. 1. S. Saito, Y. Endo and E. Hirota, J. Mol. Spectrosc. 98 (1983) 138.

32

T.J. Sears / Energy IeveIs of an asymmetric top free radical

[42] J.H. Wilkinson and C. Reinsch, Handbook for Automatic Computation, vol. II, Linear Algebra (Springer Verlag, Berlin, 1971). [43] S. Castellano and A.A. Bothner-by, J. Chem. Phys. 41 (1964) 3863. [44] T.J. Sears and A.R.W. McKellar, Mol. Phys. 49 (1983) 25. [45] D.L. Albritton, A.L. Schmeltekopf and R.N. Zare, in: Molecular Spectroscopy: Modern Research, vol. II, eds. K.N. Rao and C.W. Matthews (Academic Press, New York, 1978). [46] S. Saito, Astrophys. Lett. 178 (1972) L95. [47] H.M. Pickett and T.L. Boyd, Chem. Phys. Lett. 58 (1978) 446. [48] J.A. Austin, D.H. Levy, C.A. Gottlieb and H.E. Radford, J. Chem. Phys. 60 (1974) 207. [49] J.M. Cook, K.M. Evenson, C.J. Howard and R.F. Curl, Jr., J. Chem. Phys. 64 (1976) 1381. [50] B.J. Boland, J.M. Brown and A. Carrington, Mol. Phys. 34 (1977) 453. [51] T.J. Sears, Comput. Phys. Commun. 34 (1984) 123.