Application of a Gaussian Distribution Function To Describe Molecular

J. Phys. Chem. 1996, 100, 8645-8659

8645

Application of a Gaussian Distribution Function To Describe Molecular UV-Visible Absorption Continua. 1. Theory D. Maric´ * Institut fu¨ r Sicherheitstechnologie, P.O. Box 101564, D-50455 Ko¨ ln, Germany

J. P. Burrows Institut fu¨ r Umweltphysik der UniVersita¨ t Bremen, P.O. Box 330440, D-28334 Bremen, Germany ReceiVed: August 29, 1995; In Final Form: January 23, 1996X

A number of mathematical descriptions of UV-visible absorption continua have been compared and contrasted by using some accurate measurements of the spectra of Cl2 and BrCl recorded at 298 K. The ability of such descriptions to accurately represent the absorption, their use in deconvolution, interpolation, and extrapolation was a focus of interest. The best description of continua was found to be that obtained using semilogarithmic Gaussian distribution functions. In addition, a quantum-mechanical approach based on spectral moments was developed in a novel manner as a convenient means to analyze the fitted continua and to compute their temperature dependence. The so-called “reflection method”, which has hitherto successfully been used for this purpose, was critically reanalyzed and its theoretical basis investigated. It was shown that, when correctly applied and interpreted, the reflection method yields a sufficiently accurate description of the temperature dependence of the UV-visible absorption continua for most applications in the fields of atmospheric and combustion chemistry. Furthermore, it is simpler and less computationally demanding than any other technique and is useful for obtaining the initial values needed for the iterative procedures inherent to the more accurate quantum-mechanical methods of spectral analysis. The semilogarithmic Gaussian distribution functions may also successfully be applied to the description of structured UV-visible spectra, provided that the detail of the structure is ignored.

1. Introduction Photochemical reactions play an important role in the fields of both atmospheric and combustion chemistry. A prerequisite for understanding such reactions and quantifying their rates and yields is a detailed knowledge of the corresponding UV-visible absorption spectra and the elementary processes that determine their shape and structure. However, except for O2,1-15 Cl2,16,17 Br2,18,19 and the HO2• radical,20 knowledge of UV-visible absorption spectra of atmospherically relevant species is mostly empirical, at best recorded on a dense wavelength and temperature grid. The quantum-mechanical principles of molecular spectroscopy have been understood as a result of the pioneering works of Condon,21-23 Stueckelberg,1 Herzberg and Teller,24 Gibson et al.,16 Bayliss,18 and others. They are based on the BornOppenheimer25 approximation that the vibrational and rotational levels of an electronic state are functions of a single levelindependent potential surface. An electronic spectrum in the UV-visible region at any arbitrary temperature can be calculated, provided that the potential surfaces of the electronic states between which the transition takes place and the corresponding electronic transition moment are known as functions of the space coordinates. The Born-Oppenheimer25 approximation was theoretically justified by its propounders and hitherto experimentally confirmed on numerous examples showing, for example, that the radial potentials (i.e. one-dimensional potential surfaces) of diatomic molecules can be calculated from the pertinent spectroscopic data with an accuracy of better than 0.1 cm-1.26-30 As ab initio methods for computing potential surfaces are still in their infancy,31 accurate knowledge of the shapes of the X

Abstract published in AdVance ACS Abstracts, April 15, 1996.

S0022-3654(95)02548-2 CCC: $12.00

potential surfaces needed for the computation of the UV-visible absorption spectra must be obtained via analysis of the spectroscopic data. This means that the quantum-mechanical methods do not replace experimental work but only supplement it as a means for inter- and extrapolation of the absorption cross sections to wavelengths and temperatures other than those at which they were recorded. The computation of potential surfaces from spectroscopic data and Vice Versa requires the Schro¨dinger32 equation to be solved for every ro-vibrational level of concern. The early spectroscopic works1,16,18 relied on various approximative methods, whose accuracy is inferior to that achieved experimentally. The simplest way to solve the Schro¨dinger32 equation for any arbitrary potential surface is via the computationally demanding finite difference integration.33-44 Consequently, the widespread use of the quantum-mechanical methods has had to await the advent of modern computers. Here we are primarily concerned with the UV-visible absorption continua of diatomic and quasi-diatomic molecules (e.g. RO2• radicals) of interest in atmospheric and combustion chemistry. Except for the work on O2,1-15 Cl2,16,17 Br2,18,19 and the HO2• radical,20 there appears to be no systematic application of the quantum-mechanical methods to the description of such spectra. In principle, the UV-visible absorption cross sections can be directly converted to radial potentials and Vice Versa. However, it is convenient to have an empirical distribution function which closely approximates the shape of a typical absorption continuum. Several such functions have been hitherto proposed.45-49 In this study they are reviewed and compared in section 2, and some additional functions are proposed. Section 3 develops the quantum-mechanical method of spectral moments50 in a novel way in order to compute the © 1996 American Chemical Society

8646 J. Phys. Chem., Vol. 100, No. 21, 1996

Maric´ and Burrows

Figure 1. Best fits of the UV-visible absorption spectrum of Cl248 at 298 K using eqs 1, 4, and 7.

temperature dependence of these functions. The reflection approximation, which was previously proposed for this purpose,48,51 is critically reanalyzed and compared in section 4. Finally the results of the study are summarized in section 5. 2. Description of the Shape of the UV-Visible Absorption Continua via Gaussian Distribution Functions 2.1. Three-Parameter Distribution Functions. Four threeparameter Gaussian distribution functions have been hitherto proposed for the description of the shape of the UV-visible absorption continua:

[ ( )]

σ ) σmax exp -

ν˜ -ν˜ max

2

(1)

∆ν˜

by Goodeve and Taylor,45

[ ( )]

ν˜ ν˜ -ν˜ med exp σ ) σmed ν˜ med ∆ν˜

2

(2)

by Fink and Goodeve,46

{ [ ( )] }

(3)

{ [ ( )] }

(4)

ν˜ ν˜ σ ) σJ exp -a ln ν˜ J ν˜ J

2

by Johnston et al.,47 and

σ ) σmax exp -a ln

ν˜

ν˜ max

2

by Maric et al.48 In these equations, the indices “max” and “med” denote the maximum and the median absorption, respectively, the latter being defined via ∫ν0˜ med σ d ln(ν˜ ) ) ∫ν∞˜ σ d ln(ν˜ ) ) hBEinstein/2, where BEinstein is the Einstein med absorption coefficient.52,53 Note that the maximum and the median absorption coincide in eq 4 (i.e. ν˜ max ) ν˜ med and σmax ) σmed). Equations 1 and 2 were sporadically used45,46,54,55 until the former was published by Herzberg.56 Independently, Sulzer and Wieland51 recognized that these functions correctly describe the shape of the UV-visible absorption continua recorded at thermal equilibrium, i.e. that their applicability is not restricted to

transitions from the ground vibrational level, as previously believed. This holds implicitly for all four functions. On the basis of the reflection approximation (cf. section 4), Sulzer and Wieland51 derived a simple expression for the temperature dependence of eq 1. Consequently, eqs 1 and 2 were thereafter often used in the analysis of the UV-visible absorption continua.19,48,49,57-77 Equation 3 was hitherto solely used to describe the shape of the temperature insensitive Herzberg continuum of O2 in the range 206 e T e 327 K.15,47,78-82 In contrast, eq 4, combined with the results of Sulzer and Wieland51 (cf. section 4.5), has been successfully used to describe the shape and predict the temperature dependence of the UV-visible absorption spectra of Cl2,48 Br2,83 BrCl,77,83 CF3I,84 and various peroxy radicals in the gas85-88 and aqueous phase.89 Equations 1 and 2 provide an equally good description of the shape of the UV-visible absorption continua, as already noted by Fink and Goodeve.46 However, the best description is offered by eqs 3 and 4, which are related via σJ ) σmaxe-1/(4a) and ν˜ J ) ν˜ maxe-1/(2a), as shown by Maric et al.48 For example, Figures 1 and 2 compare the best fits of the UV-visible absorption spectra of Cl248 and BrCl83 at 298 K using eqs 1 and 4. Both spectra were deconvoluted into their constituent 1 + bands, except for the D0+ u r X ∑g band of Cl2, which was not fitted because its shape is not well-defined by the available experimental data. In this and all subsequent fits ∑(σexperimental - σfitted)2/σexperimental was minimized,90 where σ denotes the absorption cross section. Although eqs 1 and 4 both describe the principle features of the fitted spectra correctly, careful inspection of Figures 1 and 2 reveals that the goodness of the fits using eq 4 is better than that for eq 1 as measured by ∑(σexperimental - σfitted)2/σexperimental,90 the most notable deviations being indicated. The parameters obtained by fitting eq 4 to the UV-visible absorption spectra of Cl2 and BrCl (and Br2) at 298 K have been reported previously.48,83 We recommend the use of eq 4 for describing the shapes of the UV-visible absorption continua. Recently, Hubinger and Nee77 reported that eq 4 fits their UV-visible spectrum of BrCl better than eq 2 does, in agreement with our recommendation. 2.2. Multiparameter Distribution Functions. To account for the skewness, kurtosis, etc., of the UV-visible absorption

UV-Visible Absorption Continua

J. Phys. Chem., Vol. 100, No. 21, 1996 8647

Figure 2. Best fits of the UV-visible absorption spectrum of BrCl83 at 298 K using eqs 1, 4, and 7.

continua, Joens49 proposed an extended form of eq 2: 2k

(

2k

)

TABLE 1: Parameters Obtained by Fitting Eq 7 to the UV-Visible Absorption Spectra of Cl248 and BrCl83 at 298 K fit-parameters of eq 7a

σ ) ν˜ ∏e-Λiν˜ ) ν˜ exp -∑Λiν˜ i ) i

i)0

i)0

transitions

[ ( )]

ν˜

ν˜ -ν˜ i

k

σ0 exp -∑ ν˜ 1 ˜i i)1 ∆ν

Cl2

2i

(5)

Equations 1, 3, and 4 can be analogously extended in terms of ν˜ i or [ln(ν˜ )]i, e.g. 2k

{

2k

}

σ ) ∏e-Λi[ln(ν˜ )] ) exp -∑Λi[ln(ν˜ )]i ) i

i)0

i)0

{ [ ( )] } k

ν˜

i)1

ν˜ i

σ0 exp -∑ai ln

2i

(6)

Note that eqs 5 and 6 are defined in terms of an odd number of parameters. This can be most clearly seen from their representations on the right-hand side, where the parameters ν˜ i and ∆ν˜ i in eq 5 and ai and ν˜ i in eq 6 always appear together. With σ0 this gives an odd number of parameters. The three-parameter versions of eqs 5 and 6 are identical to eqs 2 and 4, respectively, so that their next higher versions contain five parameters. However, for eq 4 alternative four- and five-parameter extension are preferred:

{ [ ( )] } { | ( )| }

σ)

ν˜ -b σmed exp -a ln 1 - b/ν˜ ν˜ med-b

σ)

ν˜ -b σmed exp -a ln 1 - b/ν˜ ν˜ med-b

2

(7)

c

(8)

The denominators of eqs 7 and 8 permit analytical integration (cf. eqs 18, 19, and 27) and influence only to a minor extent the shape of the corresponding distribution curves. The goodness of such multiparameter distribution functions can be defined as being inversely proportional to the number of parameters needed to describe the shape of an UV-visible

BrCl

C Π1u r X1Σ+ g 1 + B3Π+ 0u r X Σg + r X1Σ+ D0 1

C1Π1 r X1Σ+ 1 + B3Π+ 0 rX Σ

σmed (cm2)

ν˜ med (cm-1)

a

b (cm-1)

2.196E-19b

30 360 24 940 43 940 26 850 22 580

65.17 95.83 47.41 66.03 144.2

4247 -3524 2793 5002 -9599

1.147E-20 6.108E-20 3.171E-19 1.474E-19

a Parameters obtained by fitting eq 4 to the UV-visible absorption spectra of Cl2, Br2, and BrCl at 298 K have been reported previously.48,83 b Read as 2.196 × 10-19.

absorption continuum. This means that eq 7 provides the best description of the UV-visible absorption spectra of Cl248 and BrCl83 at 298 K, as good results are achieved by adjusting only four parameters per band (i.e. σmed, a, νmed and b). This is illustrated in Figures 1 and 2, in which the best fits using eq 7 are indiscernible from the experimental absorption cross sections, as contrasted to those using eqs 1 and 4. The parameters obtained are listed in Table 1. The use of eq 8 does not result in any additional discernible improvement of the fits. In contrast, eqs 5 and 6 are defined only for an odd number of parameters, so that at least five parameters have to be fitted in order to obtain any improvement over their three-parameter parent eqs 2 and 4. However, in spite of the larger number of parameters (five as compared to four), the curves obtained in this way do not describe the shapes of the UV-visible absorption bands as closely as those obtained by fitting eq 7. Finally, it should be emphasized that multiparameter distribution functions such as eqs 7 and 8, being a sort of a “rubber ruler”, are not as suitable as their three-parameter analogues (e.g. eq 4) for fitting UV-visible absorption cross sections of low experimental accuracy. 3. Quantum-Mechanical Description of the UV-Visible Absorption Spectra According to the first-order perturbation treatment,21-24,50,91 the absorption cross section of an electric dipole transition at

8648 J. Phys. Chem., Vol. 100, No. 21, 1996

Maric´ and Burrows

zero total pressure between two bound electronic states is given by

σ(T) )

2π2ν˜

∑S ∑〈ψ′′′ |M|ψ′ 〉 × 3 hc 〈 J′ J′′

0

0

J′

V,J

V,J

〉

V′

∆(E′V,J - E′′V,J - ν˜ )〈ψ′V,J|M|ψ′′V,J〉

(9) T

whereas that of a transition between a bound lower and a dissociative upper electronic state is given by

σ(T) )

2π2ν˜ 30hc0

〈∑S ∫〈ψ′′ |M|ψ′ 〉 × J′ J′′

V,J

E,J

J′

∆(E′J - E′′V,J - ν˜ )〈ψ′E,J|M|ψ′′V,J〉 dE′J

〉

(10) T

where V and J are the vibrational and rotational quantum J′ are the rotational line intensity numbers, respectively; SJ′′ J′ ) 1 (such a normalization is factors normalized to ∑J′SJ′′ suitable for the present purpose, although it departs from that recommended by Whiting et al.92); ψ′V,J and ψ′′V,J denote the normalized bound radial wave functions of the upper and the lower electronic state, respectively, whereas ψ′E,J is the normalized dissociative radial wave function of the upper electronic state; M is the electronic transition moment, whose SI units are C m (M is a function of the internuclear separation r; in the Born-Oppenheimer25 approximation M does not depend on the quantum numbers V and J; for simplicity, the degeneracy factor (2 - δ0,Λ′δ0,Λ′′)(2S+1) is assumed to be contained in M and J′ , as required by Whiting et al.92); E′V,J, E′J, and E′′V,J are not in SJ′′ the energies of the corresponding quantum states expressed, for convenience, in the units of cm-1; ∆(E′V,J - E′′V,J - ν˜ ) and ∆(E′J - E′′V,J - ν˜ ) (cm) are distribution functions, which describe the broadening of the spectral lines (note that the units of the ∆-functions (and of the Dirac93 δ-function, cf. below) are always the reciprocal of those of the quantities on which these functions act). Equation 9 is suitable for computing the absorption cross sections of discrete spectral lines, as it has to be evaluated only once per line. In contrast, eq 10 with its infinitely dense grid of matrix elements overestimates the information content of an absorption continuum, which is always lower than that of a structured spectrum. In practice, eq 10 is evaluated for a finite number of usually equidistant grid points. In this manner it was successfully used to model the UV-visible absorption continua of Br2,19 Cl2,17 O2,2-15 and the HO2• radical.20 In comparison, earlier calculations, such as those of the Schumann-Runge continuum of O2 by Stueckelberg,1 Cl2 continuum by Gibson et al.,16 and Br2 continuum by Bayliss,18 relied on analytical expressions for the radial wave functions, which have been superseded by modern computational methods.33-44 3.1. Spectral Moments. As shown by Lax,50 provided that the ∆-functions in eqs 9 and 10 are Dirac93 δ-functions, an absorption band can be described in terms of its moments:

〈〈ν˜ i〉〉T ) ∫σ(T)ν˜ i d ln(ν˜ ) ) 2π2 30hc0

〈〈 |∑ Mψ′′V,J

J′

| 〉〉

J′ SJ′′ (H′J - E′′V,J)i Mψ′′V,J

(11) T

where H′J is the radial Hamiltonian operator of the upper electronic state. An independent derivation of eq 11 is provided in the Appendix.

Neglecting the difference between ∆- and δ-functions, eq 11 would yield results identical to those obtained via eqs 9 and 10 provided that a large (ideally infinite) number of spectral moments were to be computed and converted to an absorption band. However, for UV-visible absorption continua at thermal equilibrium the information content of 〈〈ν˜ i〉〉T decreases rapidly with the increase of the exponent i. In contrast, the information content of the matrix elements of eqs 9 and 10 is constant. Consequently, the infinite series of spectral moments can be truncated after the ith moment, and these can be converted provided that the shape of a typical UV-visible absorption band is correctly described via some analytical representation having i + 1 adjustable parameters, although other methods of conversion have also been reported.49,50,94 For example, for a truncation after the second, third, or fourth spectral moment, eqs 4, 7, or 8 can be used as analytical representations. This approach is also adept at computing the shapes of structured UV-visible absorption spectra, provided that the detail of the structure is ignored. A similar approach would be to reconstruct the UV-visible absorption band via, for example, eqs 4, 7, or 8 on the basis of three, four, or five absorption cross sections computed via eqs 9 and 10, but randomly chosen σ-values are not as representative as the spectral moments which are per definition averages over the entire spectrum. In spite of its potential advantages, the method of spectral moments has been used only occasionally.17,49,50,59-61,94-120 The zeroth spectral moment is related to the Einstein coefficient B52,53 via

〈〈ν˜ 0〉〉T ) hB(T) )

2π2 〈〈Mψ′′V,J|Mψ′′V,J〉〉T 30hc0

(12)

The higher moments, which define the position and the shape of the absorption band, are conveniently given in the form

〈〈 |∑ Mψ′′V,J

〈〈ν˜ 〉〉T i

J′

)

| 〉〉

J′ SJ′′ (H′J - E′′V,J)i Mψ′′V,J

〈〈Mψ′′V,J|Mψ′′V,J〉〉T

〈〈ν˜ 0〉〉T

T

(13)

which permits the use of relative values for the electronic transition moment M. Note that the first spectral moment is related to the oscillator strength via f ) (40mec02/e2)〈〈ν˜ 〉〉T. Now provided that the radial Hamiltonian operators of the upper and the lower electronic states are those of a simple vibrating rotator as described by Schro¨dinger,32

∂2 + U′ + J′(J′ + 1)/rˆ2 ∂rˆ2

(14)

∂2 + U′′ + J′′(J′′ + 1)/rˆ2 ∂rˆ2

(15)

H′J ) and

H′′J ) -

where -∂2/∂rˆ2 is the kinetic energy operator expressed in the units of cm-1, rˆ(cm0.5) ) 2πrx2µc0/h is a function of the internuclear separation r and the reduced mass µ, and U′ and U′′ are the radial potentials of the corresponding electronic states expressed, for convenience, in the units of cm-1 (in the BornOppenheimer25 approximation U does not depend on the quantum numbers V and J). Then according to the proposal of Joens and Bair114 to use H′′J in place of H′J,

UV-Visible Absorption Continua

〈〈ν˜ i〉〉T 〈〈ν˜ 0〉〉T

J. Phys. Chem., Vol. 100, No. 21, 1996 8649

Mean )

P ) 〈〈Mψ′′V,J|SJ′′ [H′′J + ∆U - 2J′′/rˆ2 - E′′V,J]i + Q R SJ′′ [H′′J + ∆U - E′′V,J]i + SJ′′ [H′′J + ∆U +

Variance )

2(J′′ + 1)/rˆ - E′′V,J] |Mψ′′V,J〉〉T 〈〈Mψ′′V,J|Mψ′′V,J〉〉T (16) 2

i

/

where ∆U ) U′ - U′′ is the difference between the upper and the lower radial potentials. P Q R , SJ′′ , and SJ′′ depend The rotational line intensity factors SJ′′ on the nature of the electronic transition whose assignment is unfortunately not always simple. However, as noticed by LeRoy et al.,19 the use of the Q-branch approximation, i.e. J′ ) J′′ g P R Q ) SJ′′ ) 0, and SJ′′ ) 1 for the normalization used in this 0, SJ′′ work, leads to no significant loss of accuracy (even if the 1 + Q-branch is absent, as in the Br2(B3Π+ 0u r X Σg ) transition 19 studied by LeRoy et al. ). Consequently, the above equation reduces to

〈〈ν˜ i〉〉T 〈〈ν˜ 0〉〉T

)

〈〈Mψ′′V,J|(H′′J + ∆U - E′′V,J)i|Mψ′′V,J〉〉T 〈〈Mψ′′V,J|Mψ′′V,J〉〉T

〈ν˜ 0〉 ) hBEinstein ) ∫σ d ln(ν˜ ) ) σmedxπ/a

(18)

and

〈ν˜ i〉

σ ) ∫ ν˜ i d ln(ν˜ ) ) 〈ν˜ 0〉 〈ν˜ 0〉 i

i i2/(4a)

(ν˜ med - b) e

-∑

i-k i!(-b)k 〈ν˜ 〉

k)1 k!(i-k)!

〈ν˜ 0〉

(19)

where, for convenience, the temperature dependence of the parameters of eq 7, σ(T), σmed(T), a(T), ν˜ med(T), and b(T), and that of the spectral moments, 〈〈ν˜ i〉〉T, are not explicitly indicated, but are not ignored. Note also that the oscillator strength of an absorption band described via eq 7 equals f ) (40mec02/e2)〈ν˜ 〉 ) (40mec02/e2)σmedxπ/a[(ν˜ med - b)e1/(4a) + b]. Conversely, to compute the parameters of eq 7 from the spectral moments, it is useful to define the mean, variance, and skewness of an UV-visible absorption band in terms of its moments:

(20)

〈ν˜ 0〉

〈(ν˜ - Mean)2〉 〈ν˜ 0〉

)

〈ν˜ 2〉 〈ν˜ 0〉

- Mean2

(21)

and (cf. eq 8.18 given by Lax50)

Skewness ) )

x

〈(ν˜ - Mean)3〉 〈ν˜ 0〉

(

〈(ν˜ - Mean)2〉1.5

)/

〈ν˜ 2〉 3 Mean + 2Mean3 Variance1.5 (22) 〈ν˜ 0〉 〈ν˜ 0〉 〈ν˜ 3〉

Using these definitions, the parameters of eq 7 can be computed via

(17)

Finally, minor deviations from the simple vibrating rotator model,32 such as the missing rotational levels, the influence of nuclear spin on the population of rotational levels of homonuclear molecules, Λ- and Ω-doubling, and minor perturbations, can be neglected. 3.2. Implementation of the Method of Spectral Moments. The method of spectral moments can be used in combination with the Gaussian semilogarithmic distribution functions (eqs 4, 7, and 8), provided that they correctly describe the shape of the analyzed UV-visible absorption band and that (i) their relationship to 〈〈ν˜ i〉〉T is known, (ii) a method for computing ψ′′V,J is available, (iii) a method for the evaluation of 〈〈Mψ′′V,J|(H′′J + ∆U - E′′V,J)i|Mψ′′V,J〉〉T is available, and (iv) U′′ is known and ∆U, U′′, and M are all mathematically expressed in an appropriate manner as functions of r. A description of the methods used to fulfill these requirments is given below. 3.2.1. Relationship between the Gaussian Semilogarithmic Distribution Functions and the Spectral Moments. Provided that the shape of an UV-visible absorption band can be described via eq 7, its moments can be computed via the analytical expressions

〈ν˜ 〉

σmed ) 〈ν˜ 0〉xa/π

(23)

a ) 0.5/ln[1 + Variance/(Mean - b)2]

(24)

/x1 + V

ν˜ med ) b + (Mean - b)

b ) Mean -

ariance/(Mean

- b)2 (25)

xVariance Z - 1/Z

(26)

where Z ) (Skewness/2 + x1 + Skewness2/4)1/3 (note that b is obtained by solving the cubic equation [xVariance/(Mean - b)]3 + 3[xVariance/(Mean - b)] - Skewness ) 0). All these relationships, although based on eq 7, are also valid for eq 4, in which case b ) 0. In contrast, the conversion of the coefficients of eq 8 to the spectral moments and Vice Versa must be performed numerically, except for the zeroth moment, which can be computed via

〈ν˜ 0〉 ) 2σmedΓ(1/c)/(ca1/c)

(27)

3.2.2. Computation of the Radial WaVe Functions. The radial wave functions of the lower electronic state ψ′′V,J were evaluated via a method developed by Cooley,34 which combines the Noumeroff33 finite difference integration with the iterationvariation procedure of Lo¨wdin.121 It is a shooting method, whose implementation on digital computers was pioneered by Cashion,35 Zare and Cashion,36 Zare,37 Blatt,38 Sloan,39 and others.40-44 In the present work the integration mesh consisted of 1001 equidistant grid points in the range 174 pm < r < 286 pm (∆r ) 0.11 pm). In agreement with Tellinghuisen,44 the WBK approximation proposed by Cooley34 to initialize the integration (eq 2.9 in ref 34) did not yield any detectable improvement of the results and was therefore not used. The chosen method seems to have become a standard in spite of the many hitherto proposed alternatives.44,112 For example, it is more versatile and was found to be less prone to round-off error than analytical expressions for the harmonic32 and Morse122 radial wave functions. 3.2.3. EValuation of 〈〈Mψ′′V,J|(H′′J + ∆U - E′′V,J)i|Mψ′′V,J〉〉T. As noticed by Lax50 and independently by Adler-Golden,110

〈〈ψ′′V,J|H′′J + ∆U - E′′V,J|ψ′′V,J〉〉T ) 〈∆U〈|ψ′′V,J|2〉T〉

(28)

and

〈〈ψ′′V,J|(H′′J + ∆U - E′′V,J)2|ψ′′V,j〉〉T ) 〈∆U2〈|ψ′′V,J|2〉T〉 (29)

8650 J. Phys. Chem., Vol. 100, No. 21, 1996

Maric´ and Burrows

Subsequently, Coalson and Karplus94 have shown that

〈〈ψ′′V,J|(H′′J + ∆U - E′′V,J)3|ψ′′V,J〉〉T )

〈[

∆U3 +

( )]

〈|ψ′′V,J|2〉T (30)

( )]

〈|ψ′′V,J|2〉T (31)

∂∆U ∂rˆ

2

〉

and

〈〈Mψ′′V,J|H′′J + ∆U - E′′V,J|Mψ′′V,J〉〉T )

〈[

M2∆U +

∂M ∂rˆ

2

〉

In eqs 28-30 the electronic transition moment M was assumed to be independent of the internuclear separation r (Condon21,22 approximation). Consequently, it was factored out and omitted for clarity. Unfortunately, the above and the zeroth are the only spectral moments which can be evaluated without knowledge of the derivatives of the radial wave functions of the lower electronic (k) ) ∂kψ′′V,J /∂rˆk (cf. Joens49 for 〈〈ψ′′V,J|(H′′J + ∆U state ψV,J 4 E′′ V,J) |ψ′′ V,J〉〉T). Consequently, convenient expressions of the type 〈X〈|ψ′′V,J|2〉T〉, where X is a temperature independent function, cannot be derived for any other moments. Therefore, it proved simpler to evaluate all the moments numerically,

〈〈Mψ′′V,J|(H′′J + ∆U - E′′V,J)i|Mψ′′V,J〉〉T ) 〈〈(H′′J + ∆U - E′′V,J)i/2Mψ′′V,J|(H′′J + ∆U - E′′V,J)i/2Mψ′′V,J〉〉T (32a) for even i and

) 〈〈(H′′J + ∆U - E′′V,J)(i-1)/2Mψ′′V,J| (H′′J + ∆U - E′′V,J)(i+1)/2Mψ′′V,J〉〉T (32b) for odd i, where, in a manner similar to that used in the derivation of eqs 28-31 (cf. Coalson and Karplus94), use was made of the hermiticity of the operator H′′J + ∆U - E′′V,J. The kinetic energy operator -∂2/∂rˆ2 contained in the Hamiltonian operator H′′J ) -∂2/∂rˆ2 + U′′ + J′′(J′′ + 1)/rˆ2 in eq 32 was evaluated using an approach analogous to the Noumeroff33 finite difference integration procedure used for computing the radial wave functions ψ′′V,J.34 Denoting with f the function on which the kinetic energy operator acts, it can be shown that

xj )

2fj - fj+1 - fj-1 ∆rˆ2

)

-f j(2) yj )

f j(4)∆rˆ2 12

-

rj(6)∆rˆ4 360

-

f j(8)∆rˆ6 20160

- ... (33)

14xj - xj+1 - xj-1 ) 12 -f j(2) +

f j(6)∆rˆ4 90

+

f j(8)∆rˆ6 1008

+ ... (34)

f j(8)∆rˆ6 24xj - 5yj - 2yj+1 - 2yj-1 (2) ) -f j - ... etc. zj ) 15 560 (35) where fj, xj, yj, and zj are the values of the corresponding functions at the internuclear separation rj, f ) (H′′J + ∆U E′′V,J)iMψ′′V,J, where i g 0, and f (k) ) (∂kf /∂xrˆk)j are the j derivatives of the function f at the internuclear separation rj.

Obviously, the above functions approximate the action of the kinetic energy operator -∂2f/∂rˆ2 ) -f (2) in the following order of goodness: x < y < z. However, the Noumeroff33 finite difference integration procedure used for computing the radial (6) wave functions ψ′′V,j has a residual error of - ψV,J ∆rˆ6/240 (8) 8 34 11ψV,J∆rˆ /60480 - ..., which must be also considered because f is a function of ψ′′V,J. Consequently, the total residual errors of the functions x, y, and z amount to approximately -f (4)∆rˆ2/12, f (6)∆rˆ4/144, and -f (6)∆rˆ4/240, respectively. This result was verified by comparison of the spectral moments computed via eq 32, which is based on the above functions, with those obtained via eqs 28-31, which do not rely on any such functions. The errors introduced by using function y were found to be several orders of magnitude smaller than those introduced by using function x, whereas the errors introduced by using function z were approximately -3/5 of those introduced by using function y, in agreement with theory. In addition, the predicted dependencies on ∆rˆ were found to be obeyed when the integration grid density was doubled. Therefore, only the functions x and y were used in all subsequent calculations, as the corresponding gain in accuracy does not justify the effort of computing the function z. The accuracy of this method depends in the fourth order on the mesh size of the integration grid. However, the doubling of the integration grid density did not yield any significant improvement of the results, indicating that the chosen integration mesh (cf. section 3.2.2) was sufficiently dense for the present purpose. The integral in eq 32 was replaced by a sum. Due to the hermiticity of the operator H′′J + ∆U - E′′V,J, relationships such as 〈(H′′J + ∆U - E′′V,J)Mψ′′V,J|(H′′J + ∆U - E′′V,J)Mψ′′V,J〉 ) 〈Mψ′′V,J|(H′′J + ∆U - E′′V,J)2Mψ′′V,J〉 should be satisfied exactly. Tests have shown that this is indeed so to better than 14 significant digits in double precision arithmetic, thus confirming the self-consistency of the method. Similarly, spectral moments computed via eqs 32-34 should be equal to those obtained from the Franck-Condon factors based on eqs 9 and 10, i.e. 〈Mψ′′V,J|(H′′J + ∆U - E′′V,J)i|Mψ′′V,J〉 ) ∑V′〈ψ′′V,J|M|ψ′V,J〉2(E′V,J E′′V,J)i + ∫〈ψ′′V,J|M|ψ′E,J〉2(E′J - E′′V,J)i dE′J. This equality was found to be fulfilled to better than 4 significant digits, a limiting factor being the precision of the ψ′E,J normalization. For Franck-Condon factors based solely on eq 9 and ψ′V,J bound by the outer limit of the integration grid, an agreement to better than 7 significant digits was observed. For comparison, spectral moments computed via eqs 28 and 29 should also be equal to those obtained from the Franck-Condon factors based on eqs 9 and 10, i.e. 〈∆Ui|ψ′′V,J|2〉 ) ∑V′〈ψ′′V,J|ψ′V,J〉2(E′V,J - E′′V,J)i + ∫〈ψ′′V,J|ψ′E,J〉2(E′J - E′′V,J)i dE′J for i ) 1 and 2. This relationship was found to be fulfilled to better than 4 significant digits or to better than 12 significant digits for Franck-Condon factors based solely on eq 9. Finally, note that all tested relationships assume the validity of Schro¨dinger’s32 vibrating rotator in the Q-branch approximation.19 The thermal average was computed in the usual way via

〈X′′V,J〉T ) ∑X′′V,J(2J′′ + 1)e-E′′V,J/(kT)

/∑(2J′′ + 1)e

-E′′ V,J/(kT)

(36)

where X′′V,J is the function to be averaged. It includes all spectral moments whose population densities are greater than 10-8 times that of the ground ro-vibrational level. A contraction of the sum over J′′ to several judiciously chosen representative rotational levels, as proposed by LeRoy et al.,19 was not attempted. Lowering of the truncation limit to 10-10 did not yield any detectable improvement of the results. Note also that the result does not depend on the choice of the zero-point energy E′′V)0,J)0.

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J. Phys. Chem., Vol. 100, No. 21, 1996 8651

TABLE 2: Parameters Obtained by Fitting Eq 37 to the Combined RKR Data Reported by Coxon123 for the X1Σ+ State of 79Br35Cl and 81Br35Cl parameter -1

value

parameter

value

parameter

-1

a

value

-3

β′′1 (pm ) 1.97194E-2 β′′3 (pm ) 2.46E-7 D′′e (cm ) 18244 r′′e (pm) 213.6106 β′′2 (pm-2) 3.500E-5 β′′jg4 (pm-j) 0b a Tellinghuisen.125 b Statistically insignificant and therefore set to zero.

3.2.4. Radial Potentials and the Electronic Transition Moment. The UV-visible absorption continua recorded at a single temperature usually do not contain sufficient information for the radial potentials of the upper and the lower electronic states U′ and U′′ and the radial dependence of the electronic transition moment M to be simultaneously determined. For example, if an UV-visible absorption band can be described using eq 7, this yields only four parameters σmed, a, ν˜ med, and b (cf. Table 1), which are sufficient to determine the Einstein coefficient B52,53 (cf. eq 18) and a three-parameter difference potential ∆U (cf. eq 38). Consequently, U′′ must be obtained via independent methods, whereas U′ can be derived by adding ∆U to U′′. The determination of the radial dependence of M poses a particular problem which will be discussed in section 3.3. We have chosen to demonstrate the methods of analysis of the UV-visible absorption spectra developed in this paper for 1 + the example of the BrCl(B3Π+ 0 r X Σ ) transition because the 1 + RKR radial potential of the X Σ state has been already reported.123,124 However, the reported values of U′′ are tabulated at the classical turning points, wheras the Cooley34 method for computing ψ′′V,J requires that they should be available on a grid of uniform mesh size extending to a broader range of internuclear separations than characterized by the RKR data. Therefore, the combined RKR data derived by Coxon123 for 79Br35Cl and 81Br35Cl were fitted in this work using an extended Morse122 radial potential,

{

U′′ ) D′′e 1 - exp -∑β′′j (r - r′′e ) j

[

]}

2

(37)

whose parameters β′′j and r′′e were allowed to vary, whereas D′′e was held fixed to the value reported by Tellinghuisen.125 Only the first three β′′j parameters were found to be statistically significant. The results of the fit are shown in Table 2. The largest deviation of the fitted potential from the RKR values was observed at the inner turning point of the fifth vibrational level amounting to 2.4 cm-1 or 0.10% for 79Br35Cl and -2.2 cm-1 or -0.09% for 81Br35Cl. We have no explanation for this anomalously large discrepancy. The difference potential was approximated using a threeparameter expression:

∆U ) u + Ve-w(r-re′′)

(38)

Although any other suitable three-parameter function would have performed equally well (e.g. tests using ∆U ) u + V/rw yielded similar results), the above was chosen because it is related to eq 53, which yields particularly simple results in combination with the reflection approximation (cf. section 4.5). The essence of the method of spectral moments, as used here, it to hold the number of moments and the number of adjustable parameters mutually equal. To satisfy this requirement, all higher moments are neglected. For example, starting with trial values for the parameters u, V, and w of eq 38, the first three spectral moments were computed via eqs 17, 32-34, and 36 and compared with those deconvoluted via eqs 7 and 19 from

the experimentally determined UV-visible absorption spectrum of BrCl at 298 K.83 The trial parameters were then corrected using the Newton-Raphson method,126 and the cycle was repeated until satisfactory convergence was achieved, the criterion for the convergence being the identity between the two sets of moments. The required partial derivatives ∂〈〈ν˜ i〉〉T/ [〈〈ν˜ 0〉〉T ∂(u, V, and w)] were approximated via the corresponding finite differences. Provided that an UV-visible absorption band is described via eq 4, only the first two spectral moments would need to be compared and the number of adjustable parameters in eq 38 would have to be reduced to two by omitting the parameter u. Similarly, in the case of eq 8 four spectral moments and a fourparameter description of ∆U should be used. This procedure avoids the computationally intensive method of least squares. To test its appropriateness, the fourth in this example redundant moment was also computed and found to agree to better than 5 significant digits in double precision arithmetic with that deconvoluted from the spectrum, thus confirming that it does not contain any significant information other than that already contained in the first three spectral moments providing the basis for the convergence. Procedures other than that used here are also feasible. In agreement with LeRoy et al.,19 convergence was facilitated when ∆U was expressed in terms of its derivatives at the equilibrium internuclear separation of the lower electronic state r′′e, (0) (2) (1) ∆U ) ∆Ur′′ + {exp[∆Ur′′ (r - r′′e)/∆Ur′′ ] - 1} × e e e (1) 2 (2) ) /∆Ur′′ (39) (∆Ur′′ e e (k) ) (∂k∆U/∂rk)r′′e is the derivative of ∆U at r′′e. where ∆Ur′′ e These parameters were thus used in the iterative procedure in place of u, V, and w. Finally, the electronic transition moment was defined via

M ) MeeMr(r-r′′e )

(40)

which yields particularly simple results in combination with the reflection approximation (cf. section 4.5). For small values of Mr eq 40 departs inappreciably from the usual linear approximation for the electronic transition moment (i.e. MeeMr(r-r′′e ) ≈ Me[1 + Mr(r - r′′e )]), but in contrast to the latter it is always positive. In this work tests using the linear form yielded similar results, although M behaved unphysically, changing its sign within the range of the integration grid. The evaluation of M will be discussed in section 3.3. Whereas U and M do not in the Born-Oppenheimer25 approximation depend on the reduced mass µ, the kinetic energy operator -∂2/∂rˆ2 and the centrifugal potential J(J + 1)/rˆ2 do because rˆ ) 2πrx2µc0/h, cf. eqs 14 and 15. Consequently, the energies, radial wave functions, spectral moments, and absorption cross sections all depend on µ. Instead of summing weighted spectral moments (or absorption cross sections) computed separately for each of the four naturally occurring BrCl isotopes, a single hypothetical isotope with an average reduced mass µ ) 24.55 mass units was assumed. LeRoy et al.19 quantified the effect of an analogous approximation for the natural mixture of Br2 isotopes and found it to be insignificant as compared to the uncertainty of the experimental absorption cross sections. 3.3. Spectral Analysis via the Method of Spectral Moments. For the present purpose spectral analysis can be understood to imply the search for U′, U′′, and M as functions of r, using whatever data are available. Its aim is the prediction

8652 J. Phys. Chem., Vol. 100, No. 21, 1996

Maric´ and Burrows

TABLE 3: Results of the Spectral Analysis of the BrCl(B3Π0+ r X1Σ+) Band Deconvoluted via Eq 7 difference potentialb electronic upper (cm-1 pm-k) transition momentc state molecular (0) (1) (2) potentiala rotation ∆Ur′′ ∆Ur′′ ∆Ur′′ Me (Cm) Mr (pm-1) e e e (a) (b)

yes yes

Method of Spectral Moments 22 863 -425.9 4.01 7.63E-31 0 23 494 -436.3 4.94 7.26E-31 3.63E-02

(c) and (d) (e) (f)

no no yes

Reflection Method III 22 582 -426.5 5.65 7.63E-31 0 23 422 -437.6 5.80 7.27E-31 4.92E-02 22 887 -429.1 9.77 7.60E-31 4.71E-03

a The upper state potential refers to Figures 3-6. It is the sum of U′′ (eq 37 and Table 2) and ∆U (columns 3-5 and eq 39). b The difference potential is expressed in terms of its derivatives at r′′e, cf. eq 39. c The electronic transition moment is defined via eq 40. It contains the degeneracy factor,92 cf. section 3.

of the temperature dependence of the UV-visible absorption band. In this context, the radial dependence of M poses a particular problem because it is the only function which can be determined neither from the analysis of an UV-visible absorption continuum recorded at a single temperature nor via any independent experimental technique. Therefore, in the absence of any additional information recourse has to be made to the Condon21,22 approximation that M is independent of r (i.e. Mr ) 0 in eq 40). However, provided that U′ is weakly bound, the radial dependence of M can be adjusted so that the repulsive limb of 1 + U′ coalesces into its bound part. For the BrCl(B3Π+ 0 rX Σ ) transition studied in this work the bound part of the B3Π+ 0 radial potential has been evaluated via the RKR method by Coxon123 and more recently in the form of an analytical expression by Tellinghuisen.30 The latter potential (eq 8 in ref 30) was used in all subsequent calculations, the point of coalescence being chosen to be at 229.2 pm or 18 598 cm-1. Mr in eq 40 was adjusted using the Newton-Raphson method.126 The required partial derivative ∂U′229.2 pm/∂Mr was approximated via the corresponding finite difference. Table 3 compares the results obtained in this manner with those when the radial dependence of M is neglected. An additional improvement might have been to use ∆U ) U′ - U′′ in place of eqs 38 and 39 in the range of internuclear separations for which the bound part of U′ is defined.30 However, tests have shown that this does not appreciably affect the result and was thus for convenience neglected. Figure 3 shows that the repulsive limb of the BrCl(B3Π+ 0) radial potential obtained when the radial dependence of M is neglected (potential (a)) is similar in form but is shifted to smaller internuclear separations with respect to that obtained when the radial dependence of M is accounted for (potential (b)). This large discrepancy indicates the inappropriateness of the Condon21,22 approximation in case of the BrCl(B3Π+ 0 r X1Σ+) transition. Therefore, it is surprising that the simulated 1 + BrCl(B3Π+ 0 r X Σ ) band at 180 K is not very sensitive to this approximation, as can be seen by comparison of spectra (a) and (b) in Figure 4. Consequently, the neglect of the radial dependence of M appears to have a larger effect on the repulsive limb of U′ than on the predicted temperature dependence of an UV-visible continuum. In view of the low vapor pressure of BrCl at 180 K and other difficulties,83 it would be a challenging experimental feat to record its UV-visible absorption spectrum at such a low temperature with an accuracy comparable to or better than the agreement between spectra (a) and (b) in Figure 4. Additional

1 + Figure 3. Spectral analysis of the BrCl(B3Π+ 0 r X Σ ) transition via the method of spectral moments. U′′ was taken from the literature (cf. section 3.2.4), whereas the repulsive limb of U′ was computed from the UV-visible absorption spectrum of BrCl at 298 K,83 which was deconvoluted into its constituent bands using eq 7 (cf. Figure 2 and Table 1). Potential (a) is based on the Condon21,22 approximation that M is inependent of r, whereas potential (b) was obtained by adjusting the value of Mr in eq 40 so that the repulsive limb of U′ coalesces at 229.2 pm and 18 598 cm-1 into its bound part reported by Tellinghuisen (eq 8 in ref 30). For details see Table 3.

1 + Figure 4. Shape of the BrCl(B3Π+ 0 r X Σ ) band at 180 K (the lowest temperature encountered in Earth’s atmosphere) computed via the method of spectral moments using the radial potentials shown in Figure 3 and listed in Table 3. Spectrum (b) is the best result because it accounts for the radial dependence of M, as contrasted to spectrum (a). An experimental spectrum at 180 K, to improve our understanding, must be recorded with an accuracy higher than that of these theoretical predictions.

tests have shown that both the repulsive limb of the BrCl 3 + (B3Π+ 0 ) radial potential and the simulated BrCl(B Π0 r 1 + X Σ ) band at 180 K are even less sensitive to some other common approximations, such as the neglect of molecular rotation and cruder approximations for the shape of U′′ (e.g. if

UV-Visible Absorption Continua

J. Phys. Chem., Vol. 100, No. 21, 1996 8653

U′′ were not known with an accuracy comparable to that of the BrCl(X1Σ+) radial potential used in this work123). The Q-branch approximation19 used throughout this work, which accounts for molecular rotation but neglects the splitting in the P-, Q-, and R-branches, will have an even lesser impact. Consequently, for many applications low-temperature UV-visible absorption spectra obtained from room temperature measurements coupled with a theoretical interpretation may be more accurate than those obtained experimentally. 4. Reflection Method The absorption cross section of an electric dipole transition at zero total pressure can be expressed in several ways, one of which is50,94,108

σ(T) )

2π2ν˜ 30hc0

〈〈 |∑ Mψ′′V,J

| 〉〉

J′ SJ′′ δ(H′J - E′′V,J - ν˜ ) Mψ′′V,J

J′

T

(41) where the ∆-function contained in eqs 9 and 10 has been for simplicity assumed to be a Dirac93 δ-function. An independent derivation of eq 41 is provided in the Appendix. However, eq 41 contains the radial Hamiltonian operator H′J inside the δ-function and is therefore, in contrast to eqs 9-11, computationally intractable. Now provided that (i) the radial Hamiltonian operator H′J is that of a simple vibrating rotator as described by Schro¨dinger32 (cf. eq 14) and (ii) the Q-branch approximation is valid,19 then eq 41 reduces to

σ(T) )

2π2ν˜ 30hc0

×

〈〈 | (

Mψ′′V,J δ -

)| 〉〉

∂2 + U′J - E′′V,J - ν˜ Mψ′′V,J ∂rˆ2

σ(T) )

30hc0

〈〈Mψ′′V,J|δ(U′J - E′′V,J - ν˜ )|Mψ′′V,j〉〉T )

〈

|∂U′J /∂rˆ|

〉

M2〈〈ψ′′V |δ(U′ - E′′V - ν˜ )|ψ′′V 〉〉T ) 30hc0 2

M 30hc0

〈

〉

|ψ′′V |2δν˜ )U′-E′′V |∂U′/∂rˆ|

(44) T

However, even with these simplifying assumptions the reflected |ψ′′V|2 are displaced by the amount - E′′V, as shown in Figure 5. Consequently, this approach does not offer any significant computational advantages over the method of spectral moments (cf. section 3), where molecular rotation can be similarly neglected and M assumed to be constant. Nevertheless, it has been frequently used.1,18,19,22,45,46,50,54,55,61-65,107-112,120,127-151 Most workers used method I in combination with the ground ro-vibrational level only, thus avoiding the need for thermal averaging. Where in such cases the offset due to the zero-point energy E′′V)0,J)0 has been neglected, this amounts to the use of method II (cf. section 4.2). Therefore in some studies, in which this has not been clearly documented, it is difficult to discern whether method I or II has been used. Finally, the most serious drawback of method I is that it violates the Franck-Condon principle21-23,152-154 in that the vibrational momentum is not conserved during a photon-induced electronic transition, as pointed out by Adler-Golden.110 This is because the kinetic energy operator in the δ-function was explicitly set to zero, -∂2/∂rˆ2 ) 0, which means that the absorption of a photon brings a vibrating molecule to halt. Therefore, it is not surprising that this method predicts an incorrect temperature dependence of the UV-visible absorption spectra, as noted by Joens.61 4.2. Method II. Another possibility to eliminate the kinetic energy operator is to assume that -∂2/∂rˆ2 ) E′′V,J, which yields

σ(T) )

2π2ν˜

〈〈Mψ′′V,J|δ(U′J - ν˜ )|Mψ′′V,J〉〉T ) 30hc0

〈

〉

2 2π2ν˜ |Mψ′′V,J| δν˜ )U′J

T

2 2π2ν˜ |Mψ′′V,J| δν˜ )U′J-E′′V,J

30hc0

2π2ν˜

2π2ν˜

(42)

where U′J ) U′ + J′(J′ + 1)/rˆ2 is the effective radial potential of a rotating molecule in its upper electronic state. In the next step, the kinetic energy operator -∂2/∂rˆ2 has to be eliminated to make this equation computationally tractable. According to the literature, this can be done in several ways. 4.1. Method I. The simplest way to eliminate the kinetic energy operator is to assume that -∂2/∂rˆ2 ) 0, which yields

2π2ν˜

σ(T) )

(43)

30hc0

(45) T

The geometrical interpretation of method II is similar to that of method I in that |Mψ′′V,J|2 are reflected on U′J with subsequent averaging over their thermal population densities. However, because E′′V,J is not contained in the δ-function of eq 45 (in contrast to eq 43), the displacement of the reflected |Mψ′′V,J|2 is solely due to the displacement of the effective radial potential U′J by the amount J′(J′ + 1)/rˆ2 for different rotational quantum numbers J′. Therefore, provided that the molecular rotation is neglected, the thermal averaging can be performed prior to reflection because the reflection on U′ (in contrast to U′J) does not result in any displacement of |Mψ′′V |2 (cf. Figure 5):

T

where δν˜ )U′J-E′′V,J is a dimensionless Kronecker δ-function. This result can be geometrically interpreted as a reflection of |Mψ′′V,J|2 on U′J - E′′V,J with subsequent averaging over their thermal population densities. Interestingly, the same result is obtained via eq 10 assuming that the radial wave functions of the upper electronic state are normalized Dirac93 δ-functions centered at their classical turning points ψ′E,J ) |∂U′J /∂rˆ|-0.5 δ(rˆ - rˆU′J)E′J), which is the normalized zeroth term of the Taylor expansion for ψ′E,J. For simplicity, molecular rotation is usually neglected and M assumed from Condon21,22 to be independent of r, so that it can be factored out:

|∂U′J /∂rˆ|

σ(T) )

2π2ν˜

〈〈Mψ′′V |δ(U′ - ν˜ )|Mψ′′V 〉〉T ) 30hc0 2 2 2π2ν˜ M 〈|ψ′′V | 〉Tδν˜ )U′

30hc0

|∂U′/∂rˆ|

(46)

In addition, the electronic transition moment M does not depend on temperature in the Born-Oppenheimer25 approximation and can be factored out. For simplicity, M is usually (but in contrast to method I unnecessarily) assumed to be independent of r (Condon21,22 approximation). Occasionally, the normalization coefficient |∂U′/∂rˆ|-1 and the wavenumber in the prefactor 2π2ν˜ / (30hc0) are also assumed to be constant.

8654 J. Phys. Chem., Vol. 100, No. 21, 1996

Maric´ and Burrows interpreted as a reflection of |Mψ′′V,J|2 on the difference curve between the effective radial potentials of a rotating molecule in its upper and lower electronic states, ∆U ) U′J - U′′J. The rotational contributions to U′J and U′′J are identical in the Q-branch approximation19 and are thus eliminated via subtraction, so that ∆U does not depend on J. Consequently, the reflection does not result in any displacement of |Mψ′′V,J|2 (cf. Figure 5), and therefore thermal averaging can be performed prior to reflection without need to neglect molecular rotation as in method II. The electronic transition moment M can be factored out because it does not depend on temperature in the Born-Oppenheimer25 approximation, but in contrast to method I there is no need to resort to the Condon21,22 approximation that M is independent of r. As pointed out by Adler-Golden,110 method III obeys the Franck-Condon principle21-23,152-154 in that both the position and the vibrational momentum of the nuclei are conserved during a photon-induced electronic transition. Being based on the assumption that H′′J ) E′′V,J, method III yields only the zeroth term in the binomial expansion for the spectral moments in the Q-branch approximation19 (cf. eq 17), as noted by Lax50 and independently by Adler-Golden.110

〈〈ν˜ i〉〉T 〈〈ν˜ 0〉〉T Figure 5. Reflection methods compared on the example of the BrCl2 1 + (B3Π+ 0 r X Σ ) transition. The reflected |ψ′′ V,J| are shown weighted over their thermal population densities at 298 K. In the case of method I they are displaced by the amount J′(J′ + 1)/rˆ2 - E′′V,J. In contrast, no displacement results for method II when molecular rotation is neglected (J′ ) J′′ ) 0) nor for method III in the Q-branch approximation (J′ ) J′′ g 0).19 Methods II and III interpret potential (c) computed via eqs 53 and 54 as U′ and ∆U, respectively. U′ is obtained via method III by adding ∆U to U′′ (potential (d)). Good agreement between the repulsive limb of U′ obtained via method III using eqs 53 and 54 and its bound part derived by Tellinghuisen (eq 8 in ref 30) is achieved for a nonconstant M (potential (e)). Equations 53 and 54 assume harmonic ψ′′V. Surprisingly, without this assumption the agreement is not so good (potential (f)). For details see Table 3.

As shown by Sulzer and Wieland,51 this approach is simple when using harmonic radial wave functions for the lower electronic state of a nonrotating molecule ψ′′V, as these enable analytical expressions to be derived for σ(T) (cf. section 4.5). Method II has been used because of its computational simplicity for the case where molecular rotation has been neglected.51,61,66-68 Unfortunately, most workers did not recognize its distinctness and considered it to be a simplified version of method I (cf. section 4.1). Interestingly, when method II is used for spectral analysis (neglecting molecular rotation), it yields the correct position and slope but an incorrect (too low) curvature of U′ at r′′e (cf. Figure 5 and section 4.3). In addition, method II, similarly to I, predicts an incorrect temperature dependence of the UVvisible absorption spectra, as noted by Joens.61 4.3. Method III. The third approach used to eliminate the kinetic energy operator is to assume that -∂2/∂rˆ2 + U′′J ) E′′V,J (i.e. H′′J ) E′′V,J), which yields

σ(T) )

2π2ν˜ 30hc0

〈〈Mψ′′V,J|δ(∆U - ν˜ )|Mψ′′V,J〉〉T ) 2 2 2π2ν˜ M 〈|ψ′′V,J| 〉Tδν˜ )∆U

30hc0

|∂∆U/∂rˆ|

(47)

This result, originally derived by Lax,50 can be geometrically

)

〈〈Mψ′′V,J|[∆U + (H′′J - E′′V,J)]i|Mψ′′V,J〉〉T 〈〈Mψ′′V,J|Mψ′′V,J〉〉T

)

〈∆UiM2〈|ψ′′V,J|2〉T〉 + higher binomial terms 〈M2〈|ψ′′V,J|2〉T〉

(48)

Consequently, it is quantum mechanically correct to the zeroth order (i.e. it is a semiclassical method). Provided that an UVvisible absorption band is adequately described via a threeparameter equation such as eq 4 and assuming from Condon21,22 that the radial dependence of M is negligible, spectral analysis via method III will yield results identical to those obtained via the method of spectral moments by virtue of eqs 28 and 29, which are special cases of eq 48 for i e 2 and constant M.50,110 Although method III has been pioneered by Lax as early as 1952, it has been only occasionally used.50,83,110-113,120,132,151,155-160 However, several workers used a classical modification of method III, which approximates the thermal average of the probability densities of a nonrotating molecule (e.g. crystal) via 〈|ψ′′V |2〉T ≈ e-U′′/(kT)/〈e-U′′/(kT)〉50,132,155 and that of a rotating molecule via 〈|ψ′′V,J|2〉T ≈ rˆ2e-U′′/(kT)/〈rˆ2e-U′′/(kT)〉.111-113,156-160 The only difference between methods II and III, for cases where molecular rotation is neglected, is that the former contains U′ (cf. eq 46) and the latter ∆U is the δ-function:

σ(T) )

2π2ν˜

〈〈Mψ′′V |δ(∆U - ν˜ )|Mψ′′V 〉〉T ) 30hc0 2 2 2π2ν˜ M 〈|ψ′′V | 〉Tδν˜ )∆U

30hc0

|∂∆U/∂rˆ|

(49)

As method III, in contrast to II, is quantum mechanically correct to the zeroth order (cf. eq 48), spectral analysis yields a potential which approximates ∆U more closely than U′, as can be seen in Figure 5. Consequently, the correct position and slope but a too low curvature of U′ at r′′e obtained via method II (cf. section 4.2) can be explained as follows: U′ and ∆U have the same position and slope at r′′e because the position and the slope of U′′ are both zero at equilibrium. In addition, both U′ and U′′ usually have a positive upward curvature at r′′e, so that the curvature of ∆U, being the difference of these, is lower than

UV-Visible Absorption Continua

J. Phys. Chem., Vol. 100, No. 21, 1996 8655

that of U′ alone. Therefore, ∆U closely resembles the “steep linear potential”, which has been frequently assumed to be the prerequisite for the reflection approximation to be valid. Finally, method III, similar to II, is simple when using harmonic radial wave functions for the lower electronic state of a nonrotating molecule ψ′′V, as these enable analytical expressions to be derived for σ(T) (cf. section 4.5). In all other cases it does not offer any significant computational advantages over the method of spectral moments (cf. section 3). 4.4. Method IV and Other Methods. According to Eryomin and Kuz’menko,120 an improvement over the above methods is obtained assuming that the kinetic energy operator equals -∂2/∂rˆ2 ) (E′′V,J - U′′J )/3, which yields 2

σ(T) )

2π ν˜ 30hc0

×

〈〈Mψ′′V,J|δ(∆U + 2(U′′J - E′′V,J)/3 - ν˜ )|Mψ′′V,J〉〉T )

〈

〉

2 2π2ν˜ |Mψ′′V,J| δν˜ )∆U+2(U′′J-E′′V,J)/3

30hc0

|∂(∆U + 2U′′J /3)/∂rˆ|

(50) T

In this case |Mψ′′V,J|2 are reflected on ∆U + 2(U′′J - E′′V,J)/3, resulting in their displacement by the amount 2[J′′(J′′ + 1)/rˆ2 - E′′V,J]/3 in the Q-branch approximation.19 The displacement persists by the amount - 2E′′V /3 even when molecular rotation is neglected, so that method IV is similar to method I in that thermal averaging cannot be performed prior to reflection. Consequently, method IV does not offer any significant computation advantages over the method of spectral moments (cf. section 3). Eryomin and Kuz’menko120 proposed three additional ways to eliminate the kinetic energy operator in eq 42, but these are even more complex. 4.5. Application of the Reflection Method to the Harmonic Radial Wave Functions. As shown by Ott161 and independently by Sulzer and Wieland,51 the thermal average of the harmonic probability densities is given by ∞

〈|ψ′′V |2〉T ) ∑NV|ψ′′V |2 ) V)0

x

tanh π

e-ξ

2

tanh

(51)

where ψ′′V denotes the normalized harmonic radial wave function of the lower electronic state of a nonrotating molecule; NV ) {1 - exp[- hc0ω′′e /(kT)]} exp[- hc0ω′′eV/(kT)] is the Boltzmann thermal population density of the Vth vibrational level of a harmonic oscillator; tanh ) tanh[hc0ω′′e /(2kT)]; ξ ) 2π(r r′′e ) × xµc0ω′′e /h is a dimensionless function of the internuclear separation r and the reduced mass µ (note that ξ ) 0 at r′′e ); and ω′′e is the harmonic vibrational wavenumber of the lower electronic state. Provided that the electronic transition moment is defined via eq 40, it can be shown that

M2〈|ψ′′V |2〉T ) 2

Me2eMξ

xπ

xtanh eMξ (coth-1)e-[(ξ-Mξ)-Mξ(coth-1)] 2

2

tanh

(52)

where Mξ is related to Mr in eq 40 via Mr ) 2πMξ xµc0ω′′e /h. The chosen form of the above equation is not the mathematically simplest but rather the most suitable for the present derivation. Now, provided that the difference potential is defined in a manner similar to that in eq 38,

(

)

ξ - Mξ

∆U ) b + [ν˜ med(0) - b] exp -

xa(0)

(53)

then combining eqs 49, 52, and 53 yields the temperature dependence of eq 7,

σ(T) )

{ [(

)] }

ν˜ - b σmed(T) exp -a(T) ln 1 - b/ν˜ ν˜ med(T) - b

2

(54)

2

where σmed(T) ) σmed(0)xtanh eMξ (coth-1); σmed(0) ) 2π1.5/ (30hc0) × Me2eMξ2xa(0); a(T) ) a(0) tanh; and ν˜ med(T) - b )

[ν˜ med(0) - b] e-Mξ(coth-1)/xa(0). If b were zero in eqs 53 and 54, the temperature dependence of eq 4 would have been obtained. For example, the previously reported radial potentials of Br2 and BrCl are based on eq 4 (Figures 4 and 8 in ref 83). The skewness of an UV-visible absorption band, which is given by a nonzero value of b in eq 7, is attributed in eq 53 solely to the shape of ∆U, neglecting any quantum-mechanical causes resulting from binomial terms higher than the zeroth in eq 48 or from the anharmonicity of U′′. Therefore, a nonzero value of b does not necessarily imply increased accuracy in this model. A linear form of eq 53, ∆U ) ν˜ med(0) - (ξ - Mξ)∆ν˜ (0), would have resulted in the temperature dependence of eq 2 or of eq 1 provided that, in addition, the wavenumber in the prefactor 2π2ν˜ /(30hc0) of eq 49 were assumed to be constant. Since the assumption of an exponentially shaped ∆U as given by eq 53 is more realistic than that of a linear ∆U, this explains within the validity of method III the observation made in section 2 that eqs 4 and 7 describe typical shapes of UV-visible absorption continua somewhat more accurately than eqs 1 and 2. The only difference between the above results obtained via method III and those which would have been obtained via method II is that in the latter case U′ would appear in place of ∆U in eq 53, which should then be combined with eq 46 instead of eq 49. For example, Sulzer and Wieland51 neglected the radial dependence of M (Condon21,22 approximation, Mr ) Mξ ) 0 in eqs 40, 52, and elsewhere) and assumed that U′ ) ν˜ max - ξ∆ν˜ (0) and that the wavenumber in the prefactor 2π2ν˜ / (30hc0) of eq 46 was constant and derived via method II, the temperature dependence of eq 1. However, method III is correct, as shown in section 4.3, and their result51 is actually based on a linear difference potential ∆U ) ν˜ max - ξ∆ν˜ (0), as noted by Adler-Golden.110 Finally eq 53 deserves some notice, as it implies that a nonzero value of Mξ has almost no effect on the form of ∆U (except via ν˜ med(0), which weakly depends on Mξ), but simply shifts it to a different internuclear separation. This should also hold to some degree for the method of spectral moments because it is related to method III via eq 48. The shift in ∆U reflects itself in the position of U′, as can be seen by comparison of potentials (a) and (b) in Figure 3 and (d) and (e) in Figure 5. 4.6. Spectral Analysis via Method III. Spectral analysis via method III using eqs 53 and 54 can be geometrically interpreted as the search for ∆U and M as functions of ξ, such that the thermal average of the harmonic probability densities of a nonrotating molecule 〈|ψ′′V |2〉T, when multiplied with M2, reflected on ∆U and finally multiplied with 2π2ν˜ /(30hc0), yields the analyzed UV-visible absorption band. Once ∆U and M are determined, the shape of the band can be predicted for any arbitrary temperature for which the underlying assumptions are reasonably valid. A prerequisite of this procedure is that the harmonic vibrational wavenumber ω′′e, which appears in tanh-

8656 J. Phys. Chem., Vol. 100, No. 21, 1996

1 + Figure 6. Shape of the BrCl(B3Π+ 0 r X Σ ) band at 180 K (the lowest temperature encountered in Earth’s atmosphere) computed via method III using the radial potentials shown in Figure 5 and listed in Table 3. Spectra (d) and (e) are based on the harmonic model described in section 4.5. Spectrum (d) neglects the radial dependence of M, whereas spectrum (e) accounts for it, so that they should be compared with spectra (a) and (b) in Figure 4, which are the corresponding exact results. Spectrum (f) does not rely on the harmonic model but accounts for the radial dependence of M so that it should be compared with spectra (b) and (e).

[hc0ω′′e /(2kT)], is a priori known. Here it was computed from the parameters of eq 37, which are given in Table 2, i.e. ω′′e ) β1

xD′′e h/(2π2µc0) ) 441.396 cm-1.

In practice, spectral analysis via method III comprises several steps. Firstly, the parameters σmed(T), a(T), ν˜ med(T), and b of eqs 7 and 54 are determined by fitting and where necessary deconvolution in an appropriate number of individual bands. This was demonstrated in section 2.2 for the examples of the UV-visible absorption spectra of Cl248 and BrCl83 at 298 K (cf. Figures 1 and 2 and Table 1). Secondly, provided that ω′′e is known and assuming from Condon21,22 that Mξ ) 0, these parameters can be converted to σmed(0), a(0), and ν˜ med(0), whereas b does not depend on temperature in this model. Finally, using the latter parameters, the shape of the UV-visible absorption band can be predicted via eq 54 for any arbitrary temperature for which the underlying assumptions are reasonably valid. If desired, σmed(0) can be converted to Me, but this is not necessary for predicting the temperature dependence of an UV-visible absorption band. It may be noted that in the above procedure there was no need to evaluate ∆U explicitly, as only the parameters Mξ, a(0), ν˜ med(0), and b of eq 53 have had to be evaluated but not the equation itself. However, ∆U is needed if the radial dependence of M is to be determined in a manner analogous to that described for the method of spectral moments in section 3.3. In that case, the repulsive limb of U′ was obtained by adding ∆U to U′′, the latter being defined via eq 37 using the parameters listed in Table 2. The value of Mξ was then adjusted until the repulsive limb of U′ coalesced at 229.2 pm and 18 598 cm-1 into its bound part reported by Tellinghuisen (eq 8 in ref 30), cf. Figure 5. The trial and error adjustment proved to be sufficiently rapid so that the Newton-Raphson method126 was not needed. Figures 5 and 6 show that irrespectively of whether the radial dependence of M is neglected or not, the repulsive limb of the 3 + BrCl(B3Π+ 0 ) radial potential and the simulated BrCl(B Π0 r 1 + X Σ ) band at 180 K obtained via method III (potentials and spectra (d) and (e)) are similar to those obtained via the method

Maric´ and Burrows of spectral moments and shown in Figures 3 and 4 (potentials and spectra (a) and (b)). Consequently, method III should be sufficiently accurate when the use of the method of spectral moments is not justified by the quality and quantity of experimental data or if the computational effort has to minimized. For example, if in the absence of adequate experimental information, recourse has to be made to the Condon21,22 approximation that M is independent of r (e.g. when the bound part of U′ is nonexistent or unknown), the results of method III will be as accurate as those of the method of spectral moments or even identical if based on eq 4 (cf. section 4.3). Finally, Table 3 shows that the parameters computed via method III are in close agreement with those obtained via the method of spectral moments so that they can be conveniently used as initial values for the iterative procedures inherent to the more accurate quantum-mechanical methods of spectral analysis including the method of spectral moments. The simplicity of spectral analysis via eqs 53 and 54 owes to the fact that the thermal average of the harmonic probability densities of a nonrotating molecule 〈|ψ′′V |2〉T can be expressed in an analytical form via eq 51.51,161 To use method III in combination with the radial wave functions other than the harmonic and to account for molecular rotation, 〈|ψ′′V,J|2〉T has to be computed numerically. In this work the radial wave functions were calculated via the Cooley34 method described in section 3.2.2. They are based on the potential defined via eq 37 in combination with the parameters listed in Table 2. The thermal average was computed via eq 36. It includes all ψ′′V,J whose population densities are greater than 10-8 times that of the ground ro-vibrational level. ∆U was evaluated similarly as described in section 3.2.4, but the spectral moments were computed via eq 48 neglecting all binomial terms higher than the zeroth and replacing the integrals by sums. Equations 17 and 32-34 were not used. Figure 5 shows that the repulsive limb of the BrCl(B3Π+ 0) radial potential obtained in this manner (potential (f)) does not coalesce properly into its bound part reported by Tellinghuisen.30 In contrast, potential (b) in Figure 3, which was computed under otherwise identical assumptions except that it is based on the method of spectral moments, coalesces smoothly. Consequently, the nonrectifiable kink exhibited by potential (f) must be a consequence of the neglect of binomial terms higher than the zeroth in eq 48. However, potential (e), which is also based on method III but on harmonic ψ′′V, coalesces smoothly. Hence, the neglect of binomial terms higher than the zeroth in eq 48 appears to be compensated by the use of harmonic radial wave functions. Tests have shown that the neglect of molecular rotation inherent to the harmonic model is not the primary cause 1 + of this effect because for the BrCl(B3Π+ 0 r X Σ ) transition studied in this work the influence of molecular rotation on the zeroth binomial term of eq 48 is smaller than the error due to the neglect of its higher terms. It would be interesting to investigate the validity of this result on examples of other transitions and to prove it mathematically. The goodness of an approximate method for spectral analysis is usually judged by computing an UV-visible absorption band from a predefined potential (cf. Figures 3, 4, and 5 in ref 19, Figure 2 in ref 110, and Figure 3 in ref 120). In this approach method III scores poorly and no conclusion can be drawn as to whether it yields ∆U as required by theory when applied for the analysis of a predefined UV-visible absorption band. To our knowledge only Eryomin and Kuz’menko120 compared potentials obtained in this way and found that the agreement with the exact solution increases in the order method I < method III < method IV (cf. Figure 4 in ref 120).

UV-Visible Absorption Continua The present approach is more realistic because it comprises both steps in their natural order. Firstly, an experimentally determined UV-visible absorption band is converted to a potential, and secondly, the potential determined in this manner is then used to compute the shape of the band at a different temperature. Provided that the same method is used in both steps, it effectively cancels its own inherent error, thus enabling the temperature dependence to be predicted with an accuracy comparable to or better than that of an experiment. The potential, being an intermittent result, must be plausible but does not necessarily need to be accurate because this is the price for the use of a computationally simpler method. This explains the better agreement between the predicted shapes of the 1 + BrCl(B3Π+ 0 r X Σ ) band at 180 K shown in Figures 4 and 6 as contrasted to that between the corresponding results for the repulsive limb of the BrCl(B3Π+ 0 ) radial potential shown in Figures 3 and 5. 5. Summary Several Gaussian distribution functions, i.e. eqs 1-8, some of which have long been known to approximate the shapes of UV-visible absorption continua, were assessed, and their relationship to the reflection method22 and the method of spectral moments50 was established. The various forms of the reflection method were also assessed, and the form arbitrarily named method III, which obeys the Franck-Condon principle,21-23,110,152-154 was found to be related to the method of spectral moments via eq 48.50,110 In addition, the following was established: (i) eqs 1 and 2 and their temperature dependence can be derived via method III assuming a linear difference potential ∆U ) U′ - U′′; (ii) eq 3 can be translated into the somewhat simpler eq 4;48 (iii) eqs 4 and 7 and their temperature dependence can be derived via method III assuming an exponentially shaped ∆U, as given by eq 53; (iv) the radial dependence of the electronic transition moment M can be readily accounted for in these derivations, provided that it has an exponential form given by eq 40. An additional relationship between eqs 4, 7, and 8 and the method of spectral moments was established via eqs 18-27. Provided that the spectral moments are computed via eq 48, neglecting all binomial terms higher than the zeroth, this approach yields results identical to those obtained via method III. The method of spectral moments was also elaborated in a novel way in order to compute the temperature dependence of the Gaussian distribution functions exactly. Spectral analysis was demonstrated by using the example of 1 + the BrCl(B3Π+ 0 r X Σ ) band deconvoluted via eq 7, and the results obtained via method III were compared with those obtained via the method of spectral moments. A means of determining the radial dependence of M was demonstrated. The simplest approach was shown to be via eq 54, which is based on method III. Whether the method of spectral moments or eq 54 should be used to compute the temperature dependence of the UV-visible absorption continua depends on the quality and quantity of the available experimental data and the affordable computational effort. However, even if the method of spectral moments is used, reliable initial values required by its iterative procedure are conveniently obtained via eq 54. Unfortunately, the Gaussian distribution functions are unsuitable for describing the shapes of strongly perturbed UV-visible absorption bands or those that exhibit Condon21-23 interference structure due to the presence of extremes in ∆U within the Franck-Condon region.111,112 In the case of the BrCl(B3Π+ 0 r X1Σ+) transition the perturbation of the BrCl(B3Π+ 0 ) radial potential and hence ∆U is outside the Franck-Condon region

J. Phys. Chem., Vol. 100, No. 21, 1996 8657 (an avoided crossing, cf. Figure 8 in ref 83) and has therefore negligible influence on the shape of the corresponding absorption band. In conclusion, a hitherto most convincing quantum-mechanical explanation for the success of the Gaussian distribution functions in describing the shapes of UV-visible absorption continua has been provided. In addition, the relationship of the Gaussian distribution functions to the method of spectral moments indicates that these are also suitable for describing the shapes of structured UV-visible spectra, provided that the detail of the structure is ignored. The best description is offered by eqs 4, 7, and their “rubber ruler” version, eq 8, because they do not require that ∆U should be linear. Whether eq 4, 7, or 8 should be used for the description of the UV-visible absorption spectra depends on the quality of the experimental data. Acknowledgment. We are indebted to K. Chance, R. Huie, and J. Tellinghuisen for helpful comments, J. N. Crowley for active interest in this work, V. Rozanov and R. Spurr for mathematical advice, K.-H. Mo¨bus for the fitting program, and J. von Jouanne for help in assaying the literature. This work was in part supported by the EU STEP 0012-C (EDB) and EV5V CT93-0338 and in part by the University of Bremen. The assistance of the Institut fu¨r Sicherheitstechnologie is acknowledged. Appendix: The Derivation of Eqs 11 and 41 The ∆-functions in eqs 9 and 10 do not depend on r and may consequently enter the integrals

σ(T) )

2π2ν˜ 30hc0

×

〈∑S ∑〈ψ′′ |M∆(E′ J′ J′′

J′

V,J

V,J

- E′′V,J - ν˜ )|ψ′V,J〉〈ψ′V,J|M|ψ′′V,J〉

V′

〉

T

(A-1a) σ(T) )

2π2ν˜ 30hc0

×

〈∑S ∫〈ψ′′ |M∆(E′ - E′′ - ν˜ )|ψ′ 〉〈ψ′ |M|ψ′′ 〉 dE′〉 J′ J′′

V,J

V,J

J

E,J

V,J

E,J

J

J′

T

(A-1b) Applying the relationships E′V,Jψ′V,J ) H′Jψ′V,J and E′Jψ′E,J ) H′Jψ′E,J yields

σ(T) )

2π2ν˜ 30hc0

×

〈∑S ∑〈ψ′′ |M∆(H′ - E′′ - ν˜ )|ψ′ 〉〈ψ′ |M|ψ′′ 〉〉 J′ J′′

J′

V,J

V,J

J

V′

V,J

V,J

V,J

T

(A-2a)

σ(T) )

2π2ν˜ 30hc0

×

〈∑S ∫〈ψ′′ |M∆(H′ - E′′ - ν˜ )|ψ′ 〉〈ψ′ |M|ψ′′ 〉 dE′〉 J′ J′′

J′

V,J

J

V,J

E,J

E,J

V,J

J

T

(A-2b) where the equalities ∆(E′V,J - E′′V,J - ν˜ )ψ′V,J ) ∆(H′J - E′′V,J -

8658 J. Phys. Chem., Vol. 100, No. 21, 1996

Maric´ and Burrows

ν˜ )ψ′V,J and ∆(E′J - E′′V,J - ν˜ )ψ′E,J ) ∆(H′J - E′′V,J - ν˜ )ψ′E,J can be verified via a power series expansion of the ∆-functions. Each absorption line can, in principle, have a different shape described by the corresponding ∆-function. However, to sum and integrate over the vibrational and dissociative wave functions of the upper electronic state, it must be assumed that the lines originating from the same ro-vibration level of the lower electronic state have the same shape. For convenience, however, it is assumed that all absorption lines are properly described via the Dirac93 δ-function. The summation and integration over the vibrational and dissociative wave functions of the upper electronic state then yield

σ(T) )

2π2ν˜

∑S 〈ψ′′ |Mδ(H′ - E′′ - ν˜ )M|ψ′′ 〉〉 3 hc 〈 J′ J′′

0

V,J

J

V,J

V,J

J′

0

T

(A-3) which due to the hermiticity of M, reduces to eq 41. Applying the definition of spectral moments50 to eq 41 yields

〈〈ν˜ i〉〉T ) ∫σ(T)ν˜ i d ln(ν˜ ) ) J′ 〈Mψ′′V,J ∑SJ′′ δ(H′J - E′′V,J - ν˜ ) Mψ′′V,J〉 ν˜ i dν˜ ∫ | | 〉 〈 3 hc

2π2 0

J′

0

T

(A-4) where ν˜ i depends on neither temperature nor r, so that it may enter the integral within the thermal average:

〈〈ν˜ 〉〉T ) i

2π2 30hc0

×

∫〈〈Mψ′′V,J|∑SJ′′J′ δ(H′J - E′′V,J - ν˜ )ν˜ i|Mψ′′V,J〉〉 J′

dν˜ (A-5) T

The integration over ν˜ then yields eq 11. Finally, note that all functions in the above derivations are supposed to be real. Note Added in Proof: The radial dependence of M obtained 1 + in this study for the BrCl(B3Π+ 0 r X Σ ) transition is unusually large (cf. potentials (b) and (e) in Table 3 and Figures 3 and 5), which may be due to previously unrecognized problems with the deconvolution of the UV-visible absorption spectrum of BrCl because the radial dependence of M, as used here, effectively compensates for any dislocation of the corresponding absorption band. This possibility is discussed by Tellinghuisen (J. Phys. Chem., submitted), who reports a novel deconvolution procedure. However, BrCl is at room temperature in equilibrium with its precursors, Br2 and Cl2, so that successful deconvolution of its spectrum and component bands depends on the precise knowledge of their absorption cross sections. As Tellinghuisen uses the same laboratory data as used in this and previous work,48,83 new and better experiments are required to resolve this matter conclusively. Potential problems with the deconvolution of the BrCl1 + (B3Π+ 0 r X Σ ) band do not invalidate our arguments because, firstly, Tellinghuisen’s deconvolution procedure is based on eq 4, thus confirming the usefulness of Gaussian semilogarithmic functions for describing the shapes of UV-visible absorption continua and, secondly, most of our other arguments are insensitive as to whether this band was correctly deconvoluted because it serves only as an illustrative example. The results listed in Table 3 do, however, depend on the precise knowledge of its location and shape.

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