DENSITY MATRIX TREATMENT FOR THE
SOLID HD SPECTRUM IN THE VICINITY
OF THE R1(0) FEATURE
R. M. Herman
Department of Physics, The Pennsylvania State University, 104
Davey Laboratory, University Park, PA 16802, USA
Abstract. The spectroscopic structure of the somewhat complicated R1(1) feature and its
surroundings in solid HD [1] represents a long-standing mystery. The complexity arises from the
interference between the direct transition and the surrounding Q1-R phonon band. In this
approach, we first establish a density matrix formalism to describe overlapping single transitions
of different intrinsic line widths and intensities, coupled through a static interaction. This
approach is then generalized to the interaction of a single narrow transition embedded in a band
of broad transitions. Results are compared with the spectrum of ref. [1].
Keywords: solid hydrogen deuteride, rotation-translation spectra.
PACS: 78.90+t, 33.70Jg.
INTRODUCTION
While there have been several studies of the HD R1(0) feature (v,J = 0,0 → 1,1) and
the overlapping Q1(0)R phonon band (v,J = 0,0 → 1,0, together with a single phonon
excitation) [1-3] none has successfully treated the pronounced interference features
including a dramatic dip around 5 cm-1 below and an unexpected peak around 14 cm-1
above the sharp R1(0) peak. See Fig. 1 which shows the spectrum of ref. [1] and is
quite close to that of ref. [2]. Both the sharp Q1(1) and R1(0) features in the solid are
vibrationally downshifted by 7.3 cm-1 from their free molecule counterparts, while
being spaced by 85 cm-1 as they are in the free molecule. In the present problem a
static interaction couples states which are differently stochastically perturbed. In this
regard the usual situation of collisionally-coupled line mixing is not in force, and the
author is not aware of any previous similar treatment for this type of system.
THEORETICAL CONSIDERATIONS
To attack the problem we begin with a simple phase shift analysis of the random
perturbations, then generalize the result to include other stochastic effects. Consider a
single lower state 0, and a pair of upper states 1, 2 coupled by a static perturbation V .
In the impact limit, each of the density matrix elements 10 and  20 vary with time
dependence exp i t , so that state 1 cannot easily distinguish whether it was
populated by direct excitation from state 0 or, by V , from state 2. Thus relevant


relaxation properties must reflect the excitation pathway, and not simply the two states
indicated in the specification of the density matrix element. One therefore must
d
separate 10 into a sum of a directly excited part 10 coming from the almost 100%
i
populated state 0, and an indirectly excited part 10 coming from state 2. We write
the corresponding Liouville equations (with all energies as circular frequencies)

i 10 d  10  10t 10 d  C10 exp i t 
(1)

i
i
i 10  10  v12t 10  V12 20 d  20 i

(2)
with 10 t   V11 t   V00 t coll , 12 t   V11 t   V22 t coll , and C10 
1
Ep10 .
2
and 20i . To solve Eq. (1) in the impact
Corresponding equations exist for  20 d
regime, write
t
10 d  10 d exp  i  10t   10 t  dt   
(3)






with the result
t
 10 d  iC10 exp i 10   t  i  10 t  dt  
(4)



This equation can be converted into a difference equation connecting time intervals
large compared with a collision duration, and then be directly integrated as in
conventional impact theory with the result, (when one gets back to 10 d ),


10 d  iC10 exp i t   10 / 2  i 10  10   
(5)
with  10 and 10 being the standard linewidth and shift for the 10 transition by itself,
leading to the expected Lorentz profile in the absence ofV12 . This result and similar
ones for other components, could have been obtained with the equivalent equations
(6)
 10 / 2  i 10   10d  iC10 exp i t 

12 /2  i10  10i iV12 20d  20i 
 20 / 2  i  20   20d  iC20 exp i t 
 21 /2  i 20  20i  iV2110d  10i 
(7)
(8)
(9)

with line shift parameters neglected and the stochastic effects simply represented by
effective   parameters throughout. This is often done in other relaxation analyses,
where these parameters now reflect all sources of broadening. Here 12   21, while

possibly differing from  10 and  20.
An interesting (and necessary) feature of Eqs. (6-9) is that in the event that the ’s

are equal and small, the usual results for intensities and line positions of the coupled

states 1,2 as being directly excited from the 0 state are obtainable
for all values of


V12 and  20  10 . One now proceeds by allowing state 2 to be replaced by a
continuum of states k, and on the r.h.s. of Eq. (7), one sums over all states k.
Substitution of the solution of Eq. (8) and the form for Eq. (9) into Eq. (7) and
combining with Eq. (6), one can get a closed form solution for 10 d  10i . Then




substituting that result into Eq. (9), 20i can be established. The results can be written
as a (rather complicated) function of the noninterfered phonon spectrum, the
unperturbed R1(0) Lorentzian and an interaction-perturbed R1(0) Lorentzian, together
with their (Kramers-Kronig) dispersion counterparts.

RESULTS
The results of the present treatment are shown in Fig. 1, together with the spectrum
of ref. [1]. The best guess for a smoothed, noninterfering HD phonon spectrum,
consistent with the known D2 phonon spectrum [1] is also shown. This is then used in
the calculation of the total response, as indicated above
FIGURE 1. The observed [1] total spectrum  , the proposed smoothed uninterfered phonon
spectrum inferred from that of D2 [1]     , and resulting predicted total contour    for parameters
C10= 6.0, (V1k/C10C1k) = 0.060, 10 = 1.5 and 1k = k1 = 12.
These results are preliminary, and more study on both the methodology and correct
noninterfered phonon spectrum are needed.
ACKNOWLEDGEMENT
The author is pleased to acknowledge helpful discussions with R. H. Tipping and
with J. C. Lewis.
REFERENCES
1. A. Crane, and H. P. Gush, Can. J. Phys. 44, 373-398 (1976).
2. A. R. W. McKellar, and M. J. Clouter, Can. J. Phys 68, 422-427 (1990).
3. J. D. Poll, R. H. Tipping, Sang Young Lee, Sung-ik Lee, Tae W. Noh and J. R. Gaines, Phys. Rev. B 39, 1137211377 (1989), and references therein.