Chemical
Physics
ELSEVIER
Chemical Physics 200 (1995) 23-39
Charge transfer and electron-vibron coupling in dense solid
hydrogen
Zolt~n G. Soos a,*, Jon H. Eggert b,1, Russell J. Hemley b, Michael Hanfland b,2,
Ho-kwang Mao b
a Department of Chemistry, Princeton University, Princeton, NJ 08544, USA
b Geophysical Laboratory and Center for High Pressure Research, Carnegie Institution of Washington, 5251 Broad Branch Rd. NW,
Washington, DC 20015, USA
Received 4 February 1995
Abstract
We examine the role of charge transfer (CT) interactions in dense molecular hydrogen in relation to recently observed
spectroscopic properties at megabar pressures. Specifically, we consider virtual H~-H 2 states in which an electron is
transferred to a neighbor. The Mulliken CT integral t admixes H~-H~ fluctuations of overlapping molecules. The amplitude
of the charge fluctuations is the ionicity "y, which is found in H 2 dimers and lattices with nonoverlapping valence and
conduction bands for t small compared to the CT excitation energy oJc.r. Vibronic coupling within a dimer is found using a
Herzberg-Teller expansion and gives explicit expressions for vibron shifts in Raman and IR spectra and for IR oscillator
strength due to electron-vibron coupling. We estimate linear electron-vibron coupling constants and other parameters
required to interpret the vibrational data at and above the 150-GPa transition of dense hydrogen within a CT model. We
discuss important structural implications of equal IR and Raman shifts at the transition and relate anisotropic CT processes
to the orientational state of the dense molecular solid.
1.
Introduction
Solid hydrogen is a quantum solid with many
remarkable properties and extraordinarily well-resolved spectra at ambient pressure and low temperature. At these conditions, both para-and ortho-H 2 are
amenable to detailed theoretical study, and as a
result, the solid is comparatively well understood
[1-3] at its ambient-pressure density P0 as T ~ 0 K.
Corresponding author
J Present address: Department of Physics, Pomona College,
Claremont, CA 91711, USA.
2 Present address: European Synchrotron Radiation Facility, BP
220, F-38043, Grenoble, France.
Recently developed diamond-anvil techniques have
increased the range o f static pressures to ~ 300 GPa
(3 megabar), which corresponds to a compression [4]
P / P o of the solid to over 12. These experiments
document a striking evolution o f the properties o f the
molecular solid over this density range, such as a
dramatic increase in intermolecular interactions, restriction o f molecular rotations, phase transformations, and lowering o f the optical gap. At megabar
pressures, hydrogen is predicted to form an atomic
and presumably metallic solid [5], and this prediction
has motivated experimental and theoretical studies
over the years [4].
Vibrational spectroscopy has been particularly
useful for characterizing solid hydrogen at megabar
0301-0104/95/$09.50 © 1995 Elsevier Science B.V. All rights reserved
SSDI 0301-0 104(95)001 66-2
24
Z.G. Soos et al. / Chemical Physics 200 (1995) 23-39
pressures ( > 100 GPa) [4] and raises intriguing new
issues other than the crystal structure. The intramolecular stretching mode (vibron) at ~ 4 1 5 0
cm -~, for example, can be followed with increasing
pressure by either Raman or infrared spectroscopy,
as shown in Fig. 1 at 85 and 295 K. The splitting
6om-oJR between the IR and Raman vibrons indicates increasing intermolecular interactions with
pressure, a result consistent with lower pressure measurements [6-8]. The initial Raman studies documented the unexpected softening of the vibron at 30
GPa in hydrogen and 50 GPa in deuterium [9], while
the IR vibron begins to soften [10] at 140 GPa in Fig.
1. Hydrogen evolves from a system of weakly interacting molecules at low density to one with strong
intermolecular coupling at high densities.
Spectroscopic measurements also reveal
pressure-induced phase transitions in the molecular
Density (pl po)
9
7
I
P(GPa)
181
156
151
/
I
I
4250
H2
4600
QI(J)IR
•"4"m l ~ .... i,
""
295 K
4200
3800
60
~ 5
t
90
t
t
t
120
150
180
Pressure (GPa)
K
I
210
I
I
I
4350
4450
4550
4650
Fig. 2. Representative IR absorption spectra measured at 85 K.
The spectrum at 148 GPa, which has two bands, is for phase II;
the higher pressure spectra correspond to phase Ill. The lines are
Voigt profile fits with parameter values for phase III listed in
Table 2.
4400
4000 I
148
flavenumber (cm -I)
85 K
"'~
x10
11
240
Fig. 1. Pressure dependence of the Raman and infrared vibron
frequencies (Ql(J)Raman and (Ql(J)m) showing the I-II and
II-lll phase transitions. The 295 K data (phase I) are given by the
dashed curves [4,13]. The relative densities (p0=0.0435
mol/cm 3) determined from the X-ray equation of state are shown
across the top.
solid. Current data indicate that three phases persist
into the megabar range: the rotationally disordered
phase observed at low pressure (phase I); the lowtemperature phase with orientational order or broken
symmetry (phase II), and a high-pressure phase above
150 GPa (phase III) [4]. The latter phase is characterized by a discontinuity of the vibron frequency and a
dramatic ( > 103) enhancement of the IR intensity
[11-13] shown in Fig. 2. The IR intensity is central
to our discussion below of vibrational coupling to
electronic excitations. The transition is also remarkable because the magnitude of the vibron discontinuity decreases with increasing temperature and vanishes at a critical or tricritical point [12,14]. Phases I,
II, and III meet at a triple point near 130 K [4,15].
The optical properties of the solid also change at
these pressures, although the solid is initially transparent in the visible in phase III [16-18].
These observations have prompted a number of
26
ZG. Soos et al. / Chemical Physics 200 (1995) 23-39
the anisotropy of t for ~, cr * orbitals. The representative H 2 H 2 dimers in Fig. 3 have m a x i m u m t in the
L geometry and vanishing t for X or P. Intermolecular potentials [38] for H 2 dimers show L to have the
highest energy in the ground state. CT contributions
depend sensitively on molecular orientation and lattice structure. The simplicity of H 2 is well suited for
direct estimates of t, once the phase III structure
becomes available. We follow instead the procedure
for CT complexes or DA solids and estimate t from
vibrational changes at the 150-GPa transition.
In the absence of quantitative molecular or ab
initio theory, phenomenological models are typically
applied to vibrational spectra of CT solids [31] or
conjugated polymers [39,40]. These models invoke
e - p h coupling to describe shifts relative to a reference, thereby treating in detail CT contributions to
observed vibrations. Individual D and A or their ions
are suitable references for CT crystals [31,32],
whereas empirical force fields with minimal ~-electron fluctuations are the reference for conjugated
polymers [41,42]. The discontinuity of tam and ta R
at the 150-GPa transition suggests taking phase II as
the reference. Generalizations are discussed below
for modeling the vibron shifts in phase HI. We
compare these predictions with spectroscopic data
for the dense solid, including the positions, widths,
+
+
and intensities of the infrared and Raman vibron
bands as well as dielectric measurements.
The paper is organized as follows. We study CT
in Section 2 for H 2 lattices with one molecule per
unit cell and nonoverlapping valence and conduction
bands. We obtain the ionicity 3' of the lattice, the
amplitude of the charge fluctuations H ~-H 2 in Eq.(1),
and of H 2 dimers for t small compared to cocT.
Vibronic coupling within a dimer is found in Section
3 using a Herzberg-Teller expansion to obtain explicit expressions for Raman and IR shifts and IR
intensities. In Section 4 we examine the 150-GPa
transition and estimate linear e - p h constants and
other parameters needed to explain the data within a
CT model. Section 5 deals with the structural implications of equal IR and Raman shifts at the transition. We then discuss the relation between CT processes and orientational or reorientational transitions
in the dense solid, with special attention to experimental data for the 150-GPa transition.
2. Charge transfer in H 2 lattices
Band structure calculations [19-25] of solid H 2
typically start with oriented molecules in rigid lattices (i.e., clamped nuclei). Representative structures
+
T-geometry
X-geometry
~
R
(
a
)
R m---o
P-geometry
L-geometry
l.. V
o/+~
R+
S-geometry
(c)
(d)
Fig. 3. Left: Representative configurations of H 2 H 2 dimers, including parallel (P), linear (L), slanted (S) pairs consistent with one molecule
per unit cell: the phases + + and + - indicate the phases of the bonding and antibonding orbitals, respectively, for one of several H 2+ H~
charge fluctuations in Eq. (1). Right: Closed-packed structures predicted for solid hydrogen [4]. (a) is cubic Pa3; the others are based on a
hexagonal lattice, with all molecules along the c axis in (b), in the ab plane in (c) and tilted by 0 in (d).
27
Z G. Soos et al. / Chemical Physics 200 (1995) 23-39
are shown in Fig. 3 and there are many other possibilities, even within the subset with molecular centers on an hcp lattice. While molecular pairs shown
in Fig. 3 appear in the solid, the solid-state and
dimer wavefunctions are increasingly different at
short intermolecular separations. Within single-particle approximations, the electronic Hamiltonian for
a given lattice may formally be written as
Ho=
E[E~(k)Ak+A~+Ec(k)B~Bk~], (3)
k, tr
eigenstates of H 0 for CI involving charge fluctuations.
For simplicity, we discuss lattices with one
molecule per unit cell and z nearest neighbors. The
H 2 are then parallel, but the orientation relative to
crystal axes is arbitrary; the P, S, or L dimers in Fig.
3 are possible unit cells. We first transform both
bands to a localized, or Wannier, representation [43].
+
with fermion operators At,+, Bt,,~:
Ak~=N-1/2S" e ikI,A+
._p,~,
B k+~ , = N -
I/2K
P
where Ev(k) and Ec(k) are, respectively, valence
and conduction band energies and Ak+~,B~ create
an electron with spin o- and wavevector k. The k
sums in Eq. (3) have r branches for lattices with r
molecules per unit cell. We exclude compressed
lattices with Ec(k)< Ev(k') for any k,k'. The valence band is then filled in the ground state and the
energy per site is
e o = 2 Y'~Ev( k)/N.
(4)
k
Many approximations are possible for Ev(k), ranging from tight-binding to self-consistent-field (SCF)
methods. The gap between the valence and conduction bands is ~ 10 eV at ambient pressure and the
conduction band is even wider. The gap is significantly smaller at 150 GPa, but any overlap of the
indirect gap below ~ 200 GPa (i.e., the maximum
pressure of interest here) is small [4].
In a tight-binding picture, Ev(k) and Ec(k) are
directly related to transfer integrals among the atomic
or molecular basis functions and integrals of several
eV are expected. The filled valence band IG0) of Eq.
(3) is a single determinant, optimally the SCF result.
Coulomb interactions e2/rij between electrons connect IG > to other eigenstates of H 0. The resulting
configuration interaction (CI) expansion converges,
in principle, for the ground state when there is a gap
between the valence and conduction bands. In CT
dimers, intermolecular overlap generates a Mulliken
integral similar to Eq. (1) for CI between the DA
ground state and D + A - excited state [36]. The
corresponding charge fluctuations in solid hydrogen
are adjacent virtual H~-H 2 pairs, as indicated in Eq.
(I), whose wavefunctions overlap. We will use the
z -'~
. , e ikp~+
/~po-'
p
(5)
At the pth site and low pressure, AplO) is essentially the bonding lcr orbital while Be+]0) is the
antibonding lcr* orbital. Although the orbitals
changes with pressure, they remain even and odd,
respectively, under inversion in lattices with one H e
per unit cell. The Wannier operators above are based
on the band structure (Eq. (3)) and consequently
entail the same approximations. The strict orthogonality of the Wannier orbitals for p :g p' leads to
oscillations in addition to exponential decay at distances of several lattice constants.
The generalization of the charge fluctuations in
Eq. (1) to a lattice with z nearest neighbors is
+
Z
H c v = Y'~ E
t(m)(A;~Bp+m~+Bt+,~rAt,+m~)/2.
po" m=l
(6)
The Mulliken integral (Eq. (2)) is now expressed in
terms of Wannier functions as
t(m) =
--
+
( H 2 H2
+
IBp+,,~HAp~IHzHz).
(7)
We associate Hc.rlG0) with charge fluctuations arising from Coulomb interactions that are incompletely
treated in H 0. Overlap considerations suggest neighboring sites for the most important t(m), which in
Eq. (7) then involve an electron transfer from the
bonding orbital at p to the antibonding orbital at
p + m. The magnitude of t(m) depends sensitively
on nodes in the overlap region and on molecular
orientation. Herring [44] has discussed in detail the
differences between Wannier orbitals in Eq. (7) and
atomic orbitals in Eq. (2) in connection with kinetic
Z G . Soos et al. / Chemical Physics 200 (1995) 2 3 - 3 9
28
exchange, which goes as t(m) 2, and potential (Heisenberg) exchange. We will take an average t(m)
from experiment.
Since HCT contains one-electron operators, we
can explicitly incorporate charge fluctuations into
H 0. We use the inverse of the transformation given
in Eq. (5) to obtain
We will focus on the vibrational consequences of
admixing CT states rather than the slight lowering of
the ground-state energy.
The ionicity 3' is the amplitude of the charge
fluctuations, either the average number of electrons
per site in antibonding orbitals or of holes in bonding
orbitals
HCT = Y'~T(k)( A[~Bk~+B[~Aka ),
y = (GI ~ B~',~B ~ [ G ) / N = (GI ~
(8)
ktr
k,a
Ak,~A[o.[G)/N.
k,o"
with
(16)
z
T(k) = Y'~ t(m)e ikm.
(9)
m= I
Thus Her is diagonal in k-space and the solution of
The two sums in Eq. (16) are equal due to charge
conservation and, with the aid of Eqs. (10) and (11),
lead to
Ho + HCT is obtained by a canonical transformation
similar to the Bogoliubov transformation in BCS
theory[45]
st+
k~ = cos
+ + sin q~kB~-,~
~okAk~
~:,+~= - sin •k
+
ak~ + cos q~kBff~.
T(k)/D(k),
(11)
where D(k)= E ¢ ( k ) - Ev(k) is the vertical excita-
tion at wavevector k in the first Brillouin zone. We
regain the original valence and conduction bands for
T(k) = 0, when q~k -- 0.
We now have
½E(k)( ¢~,z~:,7- ~k+~k,z)
ko"
(12)
k
The vertical excitations E(k) take into account correlated H~-H~ fluctuations
E ( k ) = [ D ( k ) e + T ( k ) * T ( k ) ] 1/2.
(13)
The new ground state corresponds to a filled valence
band
r+ 10),
[G) = II-1x ~'+
ks sk~
CA)2] -1/2
(17)
(10)
tan 2q~, =
+ ~_~[E~(k)+Ev(k)].
-D(k)/E(k)]
= 1 - [1 + 4 2 / ( e t a -
The mixing of valence and conduction states is given
by
H0 + HCT = E --
3' = N - 1 E [1
k
(14)
The second expression is the special case of an
H2H 2 dimer discussed in the next section. Since the
charge fluctuations in Eq. (1) are symmetric, all H z
remain neutral on average and no charge density
wave develops.
This completes the general analysis of CT in
lattices with one H 2 per unit cell. The generalization
to r molecules per unit cell is straightforward, but
premature in the absence of structural data at megabar
pressures. The inclusion of band overlap is possible
in principle, but requires accurate information about
Ev(k) and Ec(k) at high pressure. We will use
vibrational data to estimate the ionicity, and thus CT
contributions, instead of trying to evaluate the average over the Brillouin zone in Eq. (17). Both the
vibron frequency and intensity are sensitive to CT.
The molecules in the ground state IG > are still
strictly neutral, but the admixture y of antibonding
orbitals lowers the H 2 stretch to in the ground state
to
,o2(3") = (1 - 3 ' ) , o 2 + 3"toe
(18)
k
where 10 > is the vacuum state. The ground-state
energy gained per site is
AE = E [ E ( k ) k
D(k)]/N,
(15)
where toe < to is the excited-state frequency. The
force constant goes as o92 for harmonic potentials
and represents the curvature of the filled valence
band (Eq. (4)) with respect to R e at 3' = 0, whereas
toe is the frequency at 3' = 1 and will be taken from
Z G. Soos et a l . / C h e m i c a l Physics 200 (1995) 2 3 - 3 9
experiment. One H 2 per unit cell implies that all
molecules have identical oJ(y) and bondlengths.
In this initial study, we consider small CT integrals (Eq. (2)) between neighbors in a molecular
phase with a finite band gap. T(k)/D(k) is then
small by hypothesis. The lowest-order contributions
in T(k) to the ground-state stabilization, Eq. (15),
are
AE ¢2':N ' E T ( k ) * T ( k ) / 2 O ( k )
k
=(2D)-'
~
t(m) 2.
(19)
Ill = 0
The second approximation uses an average excitation
energy D. The k sum over T * T then reduces via
Eq. (9) to a sum over neighbors. Similarly, the
ionicity to lowest order is
1
T = N ' E T ( k ) * T ( k ) / 2 D ( k ) 2 = 2D---i •t(m)
k
2.
m
(20)
The optical gap coj inferred from dielectric measurements [16] sets a lower bound on the energy D.
Since CT transitions are usually intense [36], we
assume We.r ~ w I and further simplify to w l = D.
An average D suffices in Eq. (20) when the t(m) are
not known separately. The second-order CT corrections Eqs. (19) and (20) reduce to an integral over
the first Brillouin zone. While the crude association
of D with w~ neglects band widths, it explicitly
shows the approximations needed for a localized
representation of charge fluctuations. The mean-energy approximation in Eq. (20) for the lattice then
reduces, in lowest order, to pairwise CT with z
neighbors.
29
and both vibrational and rotational quantum numbers
of the molecule are preserved. As discussed by van
Kranendonk [1], vibrons or rotons in the solid are
linear combinations of the free molecule's excitations. Since both Raman and IR selection rules require k = 0 linear combinations in crystals, the vibron is Raman active, IR forbidden in (hypothetical)
lattices with one molecule per unit cell, whose k = 0
mode is an in-phase stretch of all molecules. The
out-of-phase stretch of the two molecules in the hcp
unit cell is IR active. Explicit analysis for hcp places
the Raman vibron to be at the bottom and the IR
vibron near the band origin [1,4].
Both WlR and ¢oR are discontinuous at the 150GPa transition (Fig. 1), and the IR intensity increases
sharply (Fig. 2). Such behavior recalls phase transitions in DA and ion-radical organic solids with
prominent CT absorption bands in the visible [31].
There, loss of molecular inversion symmetry confers
strong IR activity to ag modes [34]. Painelli and
Girlando [32] have systematically analyzed vibronic
activity in ion-radical dimers such as ( A - ) 2 or
(D+)2, CT dimers DA or D+A -, and various trimers
and regular or alternating chains, all with one active
orbital per site. They also relate different approaches
and notations, including linear response (LR) [34]
and perturbation [47] theories, the original vibronic
Mulliken theory [35], and exact diagonalization lot
dimers [48]. Herzberg-Teller ( H - T ) expansions provide a unified framework for LR studies, including
coupling of backbone stretches to w electrons in
conjugated polymers [41,42].
We consider parallel dimers S, L or P in Fig. 3
with an inversion center midway between the
molecules and generalize the H - T expansion [32] to
two active orbitals. The CT Hamiltonian, Eq. (6), for
a dimer at sites p = 1 and 2 is
He=
+
Y~[ED(RI,)AI,~AI, e+ E A(Rp)Bp,rBI,~
]
per
3. Vibronic coupling in
a 2
dimers
+Et(a~,,B2~,+B,,a~,,+h.c. ).
An H 2 molecule in the gas phase has a single
vibrational degree of freedom, a stretching mode at
4161 cm -I for J = 0 (QI(0)) and 4155 cm -1 for
J = I (Q~(1)) [46]. This symmetric (Eg) mode is
Raman active, IR forbidden. At ambient or low
pressure, intermolecular interactions in H 2 are weak
(21)
o"
The orbital energies e(Rp) depend on the bondlength
and h.c. is the Hermitian conjugate. The linear e - p h
coupling constants
gD = (OeD/OR)e,
gA = ( i ) e A / O R ) e
(22)
Z.G. Soos et al. / Chemical Physics 200 (1995) 23-39
30
are evaluated at the equilibrium bondlength R e of
H 2 and are central to the model; go + gA couple to
Raman and IR-active modes, respectively.
The solutions of H e at R = R e are the crude
adiabatic states [32] for an H - T expansion. The
electronic states are elementary and correspond to
k - - 0 and v in Eq. (10), with tan2~o in Eq. (11)
given by + 2 t / ( c A - c D ) .
The energies are
_+E ( 7 ) / 2 , with
E ( y ) = [ ( c A - % ) 2 + 4t2]1/2
(23)
in agreement with Eq. (13) for k = 0 and ~r. The
ionicity y = 2sin2q~ is given in Eq. (17). The maximum y = 1 corresponds to Eo = cA, when E ( T ) i s
due entirely to t. The bonding and antibonding levels
are doubly degenerate, each with an even and odd
orbital under inversion. Thus one-photon (CT) and
two-photon excitations of H e both occur at E(y).
The ground state IG(7)) is given by Eq. (14) on
filling the two bonding orbitals. The ground-state
energy of H e is
Eo(y ) = - 2 E ( y )
+ 2 % + 2c o.
(24)
state frequency we in Eq. (18) at y = 1. The linear
term in R+ describes the bondlength variation with
ionicity. The partial derivative of the corrected
ground-state energy with respect to R+ yields
R - R e = -- y( go + gA)/Mr°2,
(27)
where o) is the vibron frequency in the absence of
CT.
The linear Rp terms in Eq. (25) also contribute in
second order. The correction to E 0 is
- ~_,I<rIH¢vlG)IZ/E,.
r
"= - s i n 2 2 q~ [( go + g a ) 2 R2
+ ( go - gA)2 R 2 ] / 2 mE(T).
(28)
Out-of-phase R contributions now appear explicitly, as does the CT excitation energy E(y). The
frequency to(y) in Eq. (18) shifts to lower energy, as
found by taking second partial derivatives with respect to R +. The Raman and IR frequencies of an
H 2 dimer are given by Eqs. (17), (18) and (23) as
O 2+ = 60( y ) 2 -- y ( 2
- y)( gD + gA ) Z / m E ( T ) .
The H - T procedure improves E 0 and ]G) by expanding H e up to quadratic terms in the bondlengths
R~, R 2. This generates a coupling Hey between
electrons and molecular vibrations. The nuclear motions are harmonic by hypothesis, with different
frequencies and bondlengths in the ground and excited states. Since we are seeking corrections to
clamped nuclei at R e, the electronic operators of Hey
are just electrons in tr * and holes in or, as given in
Eq. (16),
Second-order energy corrections indicate first-order
corrections to wavefunctions. This mixing of vibrational and electronic states alters the vibron intensities.
First-order corrections in Hev are readily obtained
for the crude adiabatic ground state IG). The dipole
moment PCGs= (GSI PCIGS) of the corrected ground
state involves the operator
Hey = E At,,~ ap+~( RpgD + R2 e ~ / 2 )
pc= ( eR/2) ~_, ( A L AI~ + B~-~Bt~ - A ~ A2~
(29)
pc¢
o"
q- ZBp+rBpcr(RpgA + R2pe~/2).
ptr
We next introduce normal coordinates R + = (R~ _+
R2)gr-m/2, where m = ran~2 is the reduced mass,
and evaluate the ground-state expectation value of
Hey. After some algebra, we find
(GIHevIG) = "/[(go + gA ) R + v/-2-/m
+(c; +c;)(RZ +R2)/2m].
-
(25)
82 )
(30)
for separation R between the molecular centers; pccs
is given by the cross terms
2 ~ (GI pcl r)( rl Hey IG)/E r
r
= R_ eR( gD - gA ) sin22 ~0( 2 / m ) 1 / 2 / E ( y ) .
(31)
(26)
As expected, the first-order correction is linear in the
ionicity. The curvature of (GI He~ ]G) with respect to
either R+ or R_ relates c o" and cA
" to the excited
An out-of-phase stretch along R induces a dipole
moment pcGs when go ¢ gA and the IR intensity
borrowed [32] from the electronic excitation goes as
(~pcas/OR_) z. We return to the IR intensity after
Z G. Soos et al. /Chemical Physics 200 (1995)23-39
evaluating the CT transition moment to ICT - ). We
express /x in terms of the operators in Eq. (10) and
recall that the dimer has a filled valence band consisting of k = 0 and 'rr to obtain
1
leT - > =
F..,(
(32)
where the ground state IG > is given by Eq. (14).
We then expand Eq. (30) in terms of the same
operators and find
/XCT = ( C T - I / x l G ( y ) )
=
eR~/y(2
-
y)
= eRsin2•
(33)
.
The oscillator strength [49] fcr =E(T)/X~T is directly related to the ionicity and also depends on R
and E( y ).
The IR oscillator strength due to excited-state
mixing follows from Eq. (31). We have [32]
"~
2
/xT,. = (~)p.(~s/aR_) / 2 S 2
=tX~,T(gD--gA)Z/2f2
mE('y) 2,
(34)
on using Eq. (33) for /Xcr. We convert to oscillator
strengths [30,49] and use Eq. (29) to eliminate the
e - p h coupling constants. The result is
×(g2_+x)-'
31
(36)
We have a Lorentzian with full width at half maximum (fwhm) of F for g2_ >> F. The IR profiles in
Fig. 2 will be used to obtain F.
The Raman shifts in Eq. (29) vary as go + gA
and closely parallel the red shifts of ag modes in
conjugated polymers relative to butadiene [41,42].
The L or S dimer (Fig. 3) resembles butadiene, with
H 2 rather than C = C bonds. Strong e - p h coupling
accounts for the prominence of backbone modes in
the resonance Raman spectra of polymers [39,40].
Excited-state mixing and vibrational frequency shifts
also imply increased Raman scattering in dense hydrogen. But the Raman vibron is allowed even in the
absence of CT and Raman spectra measured in the
visible near the transition to phase III appear to be
far from resonance [I1]. Some resonance enhancement of the Raman vibron is observed at the highest
pressures ( > 200 GPa) [50], but further work is
needed for quantitative determination of the enhancement.
4. Application to the 150 GPa transition
.f;,:SCT['2('/)
(35)
where WeT = E ( ' y ) / h is the CT frequency and A is
the dimensionless e - p h coupling constant used in
general treatments [31,32,34,39]. The IR intensity is
thus associated with the frequency decrease due to
excited-state mixing, while the red shift of w(T) in
Eq. (18) arises from the average ionicity. In our
original estimate of CT contributions [30], we took
co instead of w ( y ) and expanded WCT/~'~ tO lowest order in the IR line width F.
Similar relations hold for the IR activity of ionradical and CT dimers of organic donors and acceptors [31]. As originally discussed by Rice [34], the IR
spectrum is related to Re o'(w) and linear e - p h
coupling to a CT band can formally be described in
terms of phenomenological broadening constants F~
for each ag mode. When w m and (.oCT are well
separated, the R e e f ( w ) profile [32] normalized to
x=co-O
= 0 can be written as
1(x)/1(0)
:{I
+(2x/F)2[1-x/2(a_+x)]2}
-'
We now consider the hypothesis that the 150 GPa
transition marks the onset of CT mixing. More precisely, we associate the vibron discontinuities at P,.
with an ionicity jump due to strong CT in phase III
and weak or negligible CT in phase II. The frequencies of tO1R and w R at pressures above the transition
are listed in Table 1, along with IR oscillator strengths
obtained from Fig. 2. The situation is reversed from
organic molecular solids with prominent CT bands
and known crystal structures. Unambiguous assignments of the electronic spectrum of solid hydrogen
have not been possible due to the breadths of overlapping bands [51]. Since the CT model deals with
the ground state, we focus on the ~ 85 K data. The
observed discontinuity at Ptr is 85 cm-~ for either
the IR or Raman vibron. We take the highest frequency vibron in phase II (Fig. l) for estimating CT
contributions at Pt,- The T dependence of the Raman
discontinuity, which vanishes above a critical temperature [12,14], suggests the 0 K jump to be only
slightly larger than 85 cm -1. We estimate the ionicity in Eq. (17), using independent data for the remaining parameters. The ionicity is the central pa-
32
ZG. Soos et al. / Chemical Physics 200 (1995) 23-39
Table 1
Excitation energies of the Raman and IR vibrons and IR vibron
oscillator strengths for dense hydrogen at pressures above the
phase III transition.IR oscillator strengths are listed separately for
two different runs
Pressure
(GPa)
toR
(cm- 1)
tom
(cm- l )
fiR X 105
155.5
158.5
161.0
162.0
165.5
168.5
171.0
174.0
176.0
180.5
186.0
189.5
193.0
198.5
202.0
206.5
210.0
216.0
3957
3943
3931
3929
3911
3897
3886
3871
3862
3843
4420
4415
4409
4408
4398
4391
4386
4375
4369
4357
4339
4327
4314
4293
4278
4261
4244
4220
0.44/0.40
0.46
0.61
0.52
0.68/0.57
0.68
0.74
0.77
0.79
1.94
0.87
0.91
0.89
0.97
1.06
1.13
1.14
1.20
rameter in the CT analysis: it appears in the
bondlength (Eq. (27)), the ground-state shift (Eq.
(18), the excited-state shift (Eq. (29)), and the IR
oscillator strength (Eq.(35)).
The coupling constants gD and gA in Eq. (22)
are the slopes of the H~- and H 2 ground states at the
H 2 bondlength R e. E0(H~-) was among the first
quantitative chemical applications of quantum theory
and its derivative at R e, gD = --7.51 e V / , ~ , can be
accurately found for an isolated ion [52]. The H~
bondlength and vibrational frequency are 1.06 ,~ and
2321 cm - l . By contrast, H 2 is unstable relative to
H 2 and an electron. The variation o f E0(H 2) around
R e is taken from scattering data or theory; either
small positive or negative gA follows from recent
studies [53]. The cr * orbital should be diffuse in the
solid, spread minimally over the 12 neighbors of an
H~-. W e consequently anticipate small gA and gD =
--7.5 e V / A . Since the H 2 force constant mw 2 in
Eq. (27) is around 30 e V / A 2, the bondlength increases as ~ 0.0033, A and the change in Eq. (27) is
small. Small R e variations with pressure are in fact
obtained either experimentally [54] or theoretically
[55]. The frequency o~e in Eq. (18) is for a state with
an electron in tr and tr *. Since we are considering
charge fluctuations from the singlet ground state, we
take w e --~ 1358 cm-1 from the metastable 1 l~u state,
where the bondlength is 1.29 ,~ [56]. The next l,~u
state has toe = 2039 cm -1 and R = 1.12 ,~ [56], and
the two states mix in the solid. The H ~- frequency o f
2300 cm-~ is an upper bound since the antibonding
orbital is then unoccupied. Finally, the optical gap
hto I = E ( 3 ' ) inferred from dielectric measurements
[18] is a lower bound on Wcr of about 4 eV or
32 000 cm-~ at the transition.
These estimates and Taylor expansion o f Eq. (18)
indicate a decrease in the vibron frequency of ~
1900T cm-~ due to mixing antibonding orbitals in
the ground state; the upper bound for toe = 0 is
yo~/2 = 22003, cm -1 . The decrease o f w R and w m
due to excited states is given by Eq. (29). The
lowest-order contribution is also ~ 19003, cm-~ for
either the IR or Raman vibron when gA = 0, gD =
--7.5 e V / A , and E ( 3 , ) = 4 eV. The 85 cm - I discontinuity is partitioning about equally between
ground- and excited-state shifts and leads to 3, =
2.2%. W e can also estimate 3, from the IR oscillator
strength, Eq. (35). The transition moment /zCT in Eq.
(33) depends on the intermolecular separation, which
is R = 1.9 A at a compression [37] P / P o "~ 10; we
obtain f o r = E(T)/xzT = 23,. From the measured IR
oscillator strengths in Table 1 of ~ 0.5 × 10 -5 at
Ptr, we find 3,= 0.7% from Eq. (35) with w ( 3 , ) J 2 _ = 40 cm - l . A small ( 1 - 2 % ) admixture o f CT
states thus rationalizes both frequency and intensity
changes at the 150-GPa transition.
The CT integral t(Ptr) now follows from Eq.
(17), with Eer = EA - ED according to Eq. (23). W e
obtain t = 0.07EcT ~ 0.3 eV for y = 1%. This is
comparable to t-~ 0 . 1 - 0 . 3 eV in i o n - r a d i c a l organic
solids at ambient pressure [36] and supports our
extension of e - p h coupling to dense hydrogen. The
estimate is clearly qualitative, since we are using
molecular parameters and vibronic coupling in a
dimer instead of the dense solid.
We turn next to the vibron data in Table 1 at
P > Ptr. LR theories describe shifts relative to experimental quantities, here the vibron frequencies in
phase II. To implement such an approach above Pt,.
requires extrapolations from lower pressure. One
way is to extend the frequencies in phase II at ~ 85
K in the field of phase III. This is not a well-con-
Z.G. Soos et al. / Chemical Physics 200 (1995) 23-39
I
I
I
I
i
I
s *
s S
140
f
s* SS
'E
o
~-~r-'-c
-O ~
~O
120
100
80
~
FREQUENCYSHIFr
O
-....
--
60
I
160
L
170
]
180
40,000 * f ,12
Atom Quadratic Fit
AtoR Linear Fit
AtOp, Temperature Scans
I
190
I
200
I
210
Pressure (GPa)
Fig. 4. Pressure dependence of the vibron frequency shift and IR
oscillator strength as determined from different fits to the experimental data (see text). The dashed lines to 165 GPa shows the
error bars on the fit obtained from temperature scans (Eq. (38)).
when the Ql(J)m in Fig. 1 is the IR feature found at
low pressure. The Raman discontinuity exceeds the
IR by 20% when the first moment of the IR spectrum in phase II is used.
The second approach to the pressure dependence
of the vibron discontinuity is based on the positive
slope of the P - T phase line, so that phase III is
entered at higher P at higher T. All temperature
dependencies are assigned to CT and the procedure
is restricted to below the triple point [4], where such
a discontinuity occurs. Another assumption is that
temperature scans are truly isobaric. Pressure is not
measured at each T, although the load was constant
(the force-generating screw on the diamond cell is
fixed), and this could introduce systematic errors.
We compare the vibron frequency A (oR in phase I or
II just above Ttr with the frequency in phase III at 85
K. We considered T scans at constant P [13,57] to
obtain
A(o R = 92 + 0.2( P - 155).
strained procedure, however, because the functional
form of the pressure dependencies in phase II is not
known. Another approach is to use nominally isobaric measurements at different temperature. The
P - T slope of the phase boundary then gives the
pressure dependence of frequency shift on entering
phase III. The drawback now is that temperature
dependencies beyond the CT model are included.
In the first approach, we fit the IR vibron in Fig.
1 to a quadratic both above and below Pt,- The phase
II fit was extrapolated to phase III and combined
with phase III data to obtain the jump A ( o l R above
the transition. The result shown in Fig. 4 is
Ato m = 83 + 1 . 7 6 ( P - 155) - 0 . 0 1 4 3 ( P - 155) 2
(37)
for P in GPa and A(OlR in c m - 1 . AS shown in Fig.
1, phase II has another IR vibron with considerably
smaller intensity [13]. If we use the first moment of
the IR spectrum in phase II, A(OlR is 17 cm -I
smaller and has essentially the same pressure dependence. The pressure dependence of (OR in Fig. 1 is
almost linear. We again extrapolate phase II data to
higher pressure to obtain the jump A(oR, in cm -l,
shown in Fig. 4
Ato R = 83.7 + 1 . 9 5 4 ( P - 1 5 5 ) .
(38)
We estimate equal discontinuities A(o R and
A(OlR
33
(39)
The two approaches to A ( o R (Eqs. (38) and (39))
give different pressure dependencies and underscore
the difficulties in determining this parameter. We
must view these estimates as bounds on A(o(p) for
P > Ptr = 155 GPa, as shown in Fig. 4, that lead to
d A ( o ( P ) / d P to be between 0 and 2 c m - J / G P a .
In the context of CT at 0 K, we expect increased
intermolecular overlap and smaller band gap at higher
pressure (e.g., Ref. [18]). Both increase the ionicity
and larger T in turn increases vibron shifts for the
ground (Eq. (18)) and excited state (Eq. (29)) and
also increases the IR oscillator strength (Eq. (35)).
To lowest order in % the shifts are linear and f i R is
quadratic. The measured fm increases above Pt,.
(Table 1); this change reflects almost exclusively the
increase in phase III in view of the orders of magnitude weaker intensity below Ptr" Complete conversion of the sample to phase III at Ptr then implies
that ~
should scale as the frequency jump A (OIR"
The predicted pressure dependence is found in Fig.
4, where ~
is scaled to the IR discontinuity at 155
GPa; the effects of two-phase coexistence and pressure gradients are discussed below. Just as the CT
model gives independent estimates for Y at the transition, it implies a general relation between the shift
and intensity of the IR vibron that does not require
detailed knowledge of microscopic parameters.
34
ZG. Soos et al. / Chemical Physics 200 (1995) 23-39
We conclude the CT analysis by modeling the IR
absorption in Fig. 2. The measured linewidths (fwhm)
are listed in Table 2 for two separate runs at ~ 85 K.
Coupling to a high-energy CT state leads to the
absorption l ( x ) in Eq. (36), where x = to - O_ and
F is the fwhm of a Lorentzian. Gaussian profiles are
instead expected from instrumental effects, pressure
gradients, or other inhomogeneous broadening. We
have carried out nonlinear least-square fits for IR
lineshapes shown in Fig. 2. The observed profiles are
reasonably symmetric for phase III, although a small
correction to the base line is needed, and the lineshapes are very well fit with a Voigt function [58].
Table 2 gives the fwhm of the Lorentzian and Gaussian curves extracted from fitting Voigt profiles to the
IR absorption. The Lorentzian component is larger
( ~ 30 cm- i ) and independent of pressure up to 190
GPa; it broadens to ~ 50 cm -~ at 210 GPa.
Both the total width and its Gaussian component
in Table 2 have a dip around 170 GPa. The dip
appears in data taken on two separate samples measured three months apart, during both heating and
cooling [13]. Pressure gradients are probably a major
source of Gaussian broadening and imply frequency
shifts as discussed above. The coexistence region of
5-10 GPa for phase I or II and III around Ptr may
be due to pressure gradients. Gradual conversion to
phase III affects the IR intensity in ways not consid-
ered in the present CT analysis. We speculate that
recrystallization reduces pressure gradients immediately following the completion of the transition from
phase II to HI. No corresponding dip is observed in
the predicted Lorentzian width arising from CT
interactions. We note that overlapping ag vibrations
in typical organic solids preclude detailed fits of
band tails shown in Fig. 2.
According to the CT model, the Lorentzian width
F reflects the strength of electron-vibron coupling
to the postulated CT states. In contrast to many ag
modes in organic solids, we have a single vibration
and hence a single dimensionless e-ph coupling
constant A defined in Eq. (35). In LR theory [31,32]
A is obtained experimentally from the observed and
bare frequencies, which for dense hydrogen are O_
(Eq. (29)) and to(~/) (Eq. (18)). The electronic excitation is shifted to higher energy,
,,4, -
to ,
Ato( )2[ to( )2 _
[to(r)2_
]2
+ & r2'
(40)
and the usual case in Eq. (40) is a sum over coupled
modes on the right hand side. For a single mode, we
use Eq. (35) for A and rearrange to obtain
2 2
tour = a 2 + [ t o ( 7 ) 2 - a _ ] / r 2
(41)
Table 2
Full width at half maximum (fwhm) of the IR vibron above the transition for two separate runs. The Lorentzian ( F ) and Gaussian (A)
contributions are based on Voigt profile fits to the spectra
Pressure (GPa)
fwhm ( c m - l )
F ( c m - l)
A (cm- t)
155.5
158.5
161.0
162.0
165.5
168.5
171.0
174.0
176.0
180.5
186.0
188.5
192.0
200.0
203.5
207.0
211.5
44.3/44.0
43.7
37'.6
44.3
32.8/43.7
36.5
31.8
34.3
33.7
36.8
40.9
39.4
45.6
49.1
49.4
52.4
57.9
34.6/31.4
30.8
30.6
30.5
29.1/30.4
27.4
29.2
27.4
29.3
26.9
27.7
28.2
32.4
38.2
39.0
42.4
49.7
20.2/'23.0
23.1
15.7
24.0
10.7/23.5
17.7
8.7
14.9
11.8
18.5
22.7
20.5
23.9
22.4
22.1
22.3
21.1
ZG. Soos et al. / Chemical Physics 200 (1995) 23-39
Pressure (GPa)
50
100 150 200
20
I
16
I
I
I
5. Structural implications and CT in quantum
lattices
I
Hydrogen
°
o
4
•
Hemleyet al. (1991)
"~
Eggert et at. (1990)
• Mao & H e m l e y (1989)
[] van Straaten & Silvera (1988)
0
1
5
I
I
7
9
Density (P/Po)
35
~
11
Fig. 5. The density and pressure dependence of the effective
single oscillator toI determined from fits to index of refraction for
dense hydrogen. The thin and dotted lines are linear fits to the
data. The thicker line at 150-200 GPa shows the result assuming
COOTvaries as I/.f~m. The datum from Mao and Hemley (1989)
corresponds to the onset of proposed resonance enhancement of
the Raman vibron [52].
We estimate tom = 1.6 eV for F = 30 cm-1 when
half of the 85 c m - t discontinuity is due to excitedstate coupling. While there are no adjustable parameters, Wcv is very sensitive to the linewidth analysis
and to the partitioning of the vibron discontinuity.
Moreover, Eq. (40) describes the shift of COcr due to
e - p h coupling relative to Wcr itself. It consequently
does not include P or T dependencies of coot,
which increases in Eq. (41) with P at constant F
because the vibron shift increases.
Fig. 5 shows the results of dielectric fits to index
of refraction measurements of hydrogen as a function of density [18,59-62]. The parameter to 1 is the
frequency of a single oscillator and is expected to be
close to the direct band gap (or absorption threshold)
[63]. The magnitude of to 1 obtained in these earlier
analyses exceeds the CT estimate above. The thicker
line in Fig. 5 is that calculated assuming the COcr
varies as l / ~
[30]. The pressure dependencies of
fcv and A probably cannot be neglected, however,
and the good agreement for the slope of tOCT and o,)1
is not implied by the present model. On the other
hand, we expect on general grounds tOcr and coj to
have the same pressure dependence to lowest order.
We have rationalized features of both IR and
Raman spectra in phase III of hydrogen by assuming
that CT turns on at the transition. Other measurements indicate that molecules orientationally order in
phase III and that ordering increases with pressure
[4]. Equal IR and Raman shifts have additional implications. The dimer model (Eq. (23)) is related to
the lattices discussed in Section 2. As noted there,
the Raman-active H 2 mode involves in-phase
stretches of the entire structure, while IR activity
requires out-of-phase vibrations of two or more H e
in the unit cell. Thus CT interactions with all neighbors contribute to A WR, but only CT between crystallographically inequivalent sites contributes A W~R.
Equal IR and Raman shifts then point to strong CT
between inequivalent molecules in the unit cell and
negligible CT with equivalent neighbors. Yet the
high coordination z = 12 in close-packed lattices
suggests extensive intermolecular overlap with many
neighbors. We consequently examine the CT integral
and quantum lattices more closely.
We assume pairwise CT interactions in a closepacked lattice and emphasize that the crystal structure of hydrogen above 50 GPa is not known. However, an underlying hcp lattice for the molecular
centers is a useful starting point and is consistent
with available spectroscopic data [4]. The molecules
have z - 12 neighbors for the ab plane and c axis
shown in Fig. 6. Inequivalent sites in adjacent abab...
planes lead to the z)_~-zag chain in Fig. 6b, with
angle a = sin-i 1 / f 3 between the intermolecular
and c axes. These chains lack the inversion symmetry produced by abcabc.., layers in an fcc lattice,
since the c sites indicated in Fig. 6 are empty.
Within an hcp lattice, we must specify the orientation of one molecule in both a and b planes. We use
spherical polar coordinates 0 and q~. Symmetry precludes [64] IR activity when all molecules are along
the c axis ( 0 = q ~ = 0, structure (b) in Fig. 3), but
other orientations are consistent with an IR vibron.
Lattices with several molecular orientations in each
a or b plane have larger unit cells than hcp even if
the molecular centers form an hcp lattice.
A number of different calculations predict low-energy structures at high pressure in which the
36
Z G. Soos et al. / Chemical Physics 200 (1995) 23-39
°
Iy
jt
~',
,
,
b
b
,/
b)
Z
/ t
",
Z
/
i
r
t
f .......t
a
sites in ab p l a n e
Q
b
Fig. 6. (a) Molecular centers at a sites of an ab plane of a
hexagonal close-packed lattice; the arrow indicates the nucleus
above the plane. (b) zig-zagchain along the c axis for the oriented
molecules in (a) in successive ab planes; the empty c sites are
also shown.
molecules are uniformly tilted by 0 = 55-60 ° from
the c axis [21,23,24,65]. Energy variations in ~o are
much smaller, with minima for bisecting neighbors
in the ab plane as indicated in Fig. 6a and in
structures (c) and (d) of Fig. 3; we have 0 = 0 in (c),
0 4:0 in (d), and q~= "rr/2 in both. Similar energies
are found for different q~ in a and b planes [65], or
in structures with different q~ in the same plane [24].
If we neglect the q~ dependence, the 0 tilts correspond to axial symmetry. Any H z then participates
in three abab chains in Fig. 6b. The three chains
become inequivalent on choosing a particular tilt,
such as the yz plane, and all nuclei in Fig. 6b are in
that plane. An hcp lattice and molecular tilts based
on total-energy calculations produce a zig-zag chain
of S dimers with different CT integrals t between
successive dimers.
The anisotropy of t now becomes important. The
different symmetry of the ~r and ~ * orbitals in Eq.
(2) leads to vanishing CT for the P or X geometries
in Fig. 3. Maximum CT occurs in the L geometry,
with small internuclear spacing R23 between the
nearest protons and large spacing R]4 between the
terminal protons. We have R13 = Rz4 in S or L
dimers and contributions to t from these atomic
orbitals cancel. For parallel dimers, t increases as
R I 4 - R23 increases. As noted in connection in Eq.
(7), the magnitude of t depends on the oscillations of
the Wannier orbitals in the overlap region of the
dense solid and direct evaluation of t has been
difficult in organic CT crystals. We take t from
experiment and its anisotropy from the overlap of tr
and tr * at adjacent sites.
Although the molecular centers in Fig. 6 form an
hcp lattice with 12-fold coordination, there is a wide
distribution of CT integrals. Two types of S dimers
occur in Fig. 6b, with angles 0 - c ~ and 0 + a
between the bond and the intermolecular axes, respectively. One dimer is within 30 ° of L and has
potentially large t, while the other is almost a P
dimer with t = 0. Molecular orientation in the hcp
structure thus generates pairwise CT within dimers,
consistent with equal vibron discontinuities in the
Raman and IR. Although we have only discussed
two of the 12 neighbors in Fig. 6a, the variations in
Rq for the others are readily found: there are two P
pairs with t = 0 in the ab plane, four other equivalent neighbors in the plane, and two pairs of equivalent neighbors in the adjacent ab planes; the variations of R U are smaller, since the axis about which
the molecules are rotated is not normal to the intermolecular vector in these cases. Strong CT with a
single partner thus appears naturally in hcp lattices
of oriented molecules.
The quantum nature of solid hydrogen at ambient
pressure has no parallel in organic solids. In p-H 2,
the nuclear rotations are restricted to even angular
momenta J, as shown by roton spectra [1,2]. In the
absence of intermolecular overlap, the general expansion [66] of intermolecular potentials contains
only even J. Since the electronic ground state of H~is odd under inversion, however, its rotational function must have odd J in order to mix ICT + ) in Eq.
(1) with the ground state. Symmetry considerations
thus exclude CT in a J = 0 solid of freely rotating
H z molecules, a regime that extends to close to 100
GPa for H 2 (see Ref. [4]). Spectroscopic data suggest
that phase II is partially ordered, but neither the
structure nor possible ordering schemes have been
determined [4,13,14]. The assumption of partial ordering is consistent with the CT model proposed
here: jumping or tunneling of molecules among several equivalent minima in the structure effectively
reduces CT mixing due to the anisotropy of t. A
sharp increase ot t is possible at Pt~ on ordering the
molecules into, for example, the zig-zag chains in
Fig. 6b; t subsequently increases exponentially with
pressure as intermolecular separations decrease.
Z G. Soos et al. / Chemical Physics 200 (1995) 23-39
6. Discussion
We now consider several general features of CT
interactions and vibronic coupling in dense hydrogen. We first note that the present model is consistent with general symmetry considerations. The observed transitions may be viewed as symmetry lowering of phase I with increasing pressure [4]. Symmetry breaking arises from the loss of spherical symmetry associated with the J = 0 molecules (and rotational disorder for higher J), crystallographic distortions (also related to orientational order), and electronic effects (such as CT). In general, structural
symmetry breaking induces changes in spectroscopic
properties: normally silent vibrational modes can
become IR-active, even in homonuclear diatomic
lattices [64]. But symmetry provides no information
on the origin of the absorption (or scattering) crosssections, and that has been one of the aims of the
present study. The IR intensities in phase III are
comparable to those of ag modes in organic crystals
and have been discussed [31,32] by similar e - p h
coupling to excited states.
On a qualitative level, CT contributions may already appear below Ptr' CT effects were first proposed as an explanation for the softening of the
Raman vibron [9]. The pressure dependencies of WIR
and to R in phase 1 have been reasonably fit to ~ 60
GPa with quadrupolar, exchange repulsion, and other
intermolecular forces between nonoverlapping
molecules [66]. Quantitative description of intermolecular forces becomes difficult at higher pressure. The large increase in Fig. 1 of OJIR--WR from 3
cm --r at ambient pressure to 510 cm -1 at 180 GPa
indicates much stronger intermolecular interactions
[4,10]. The softening of IR vibron above 140 GPa
also indicates a weaker bond, possibly due to CT.
The symmetry breaking associated with phase II
could also produce weak intermolecular CT, but they
are not readily addressed within our phenomenological model since defining a reference requires accurate knowledge of all other intermolecular contributions. The pressure dependencies of O.)IR--O)R coincide at 85 and 295 K and increase smoothly through
the 150-GPa transition [4]. The equal IR and Raman
shifts for CT within H 2 dimers is consistent with the
~OIR--~OR behavior. Increased ionicity in phase IlI
softens the vibrons equally when e - p h coupling
37
primarily involves crystallographically inequivalent
molecules.
We have neglected the finite widths of bands as
well as overlap between valence and conduction
bands. Recent electronic structure calculations [23,25]
predict wide bands and a very small (indeed, in
many cases, a vanishing) band gap in phase III. The
existence of an indirect gap depends sensitively on
molecular orientation and resembles in this sense the
strong anisotropy of t. The full band structure must
then be considered and the approximations based on
small T ( k ) in Eq. (9) fail. More quantitative evaluations of CT contributions become possible once the
band structure and nearest-neighbor separations are
known, as discussed in Section 2. Qualitative aspects
of CT and increasing ionicity survive, however, as
long as the vibron frequencies remain within ~ 10%
of the molecular value. Far greater changes in vibron
frequencies are calculated in orientationally ordered,
semimetallic phase III [25,67].
In view of the large difference in zero-point energy between H 2 and D 2, a key experimental constraint is provided by isotope effects on various
physical properties. The importance of zero-point
motion has been emphasized by several groups (e.g.,
Refs. [21,26]), and there has been some effort to
incorporate this in total energy calculations (e.g.,
Ref. [25]). CT integrals are particularly sensitive to
modulation of interatomic separations, with terms
having the smallest Rij the most important. The
reduced zero-point amplitude of D 2 compared to H 2
consequently is expected to reduce CT interactions
in the heavier isotope for a given crystal structure at
the same number density; this is consistent with the
higher Pt, for deuterium [4]. There are differences as
well as similarities between the two isotopes. The IR
spectrum for phase II of deuterium [68], for example,
has up to four resolved features rather than the two
features observed for hydrogen (Fig. 1) [4,13]. The
nature of the partially ordered phase II goes beyond
vibronic analysis at Ptr and any complete model
must address isotopic differences.
We also consider the relative contributions of
inter and intramolecular CT. Intramolecular CT
would obtain if, for example, the atoms within a
molecule become crystallographically distinct. Baranowski [29] has proposed a radical symmetry breaking involving intramolecular CT: i.e., the formation
38
Z.G. Sots et al. / Chemical Physics 200 (1995) 23-39
of a true ionic state consisting of H+H - species.
Such a change is expected to generate strong dipole
absorption and vibron softening. Since the observed
vibron discontinuity represents only a ~ 3% change
in o.)R or o.)IR, the formation of a true ionic state is
unlikely in phase III. On the other hand, some
intramolecular CT cannot be ruled out and it is
important to examine both intra and intermolecular
CT for phases II and III. Moreover, we should
consider whether CT effects represent manifestations
of more subtle physics. This question is motivated in
part by theoretical predictions, so far largely untested,
that hydrogen at megabar pressures could exhibit
novel physical properties (see Ref. [4]). One possibility is exciton condensation, which may be expected
on general grounds as the band gap closes [12] and
may lead to a charge density wave. With strong
e-ph coupling, charge fluctuations may produce local distortions of the lattice (polarons) and provide a
means for the system to remain insulating to the
highest available pressures. A crucial distinction to
be made in each of these proposals is whether symmetry breaking represents a static, frozen-in structural change or is instead a dynamic or fluctuation
phenomenon, as in the CT model discussed here.
7. Conclusions
We have touched on general features of charge
transfer and vibronic coupling in hydrogen at high
densities. A simple 0 K model of CT in H 2 dimers
captures the main vibrational features of the 150-GPa
transition and leads to surprisingly specific structural
implications for phase III. Vibrational data provide
important constraints on proposed structures and ordering schemes. On general grounds, we expect CT
to become increasingly important with density as the
system evolves toward a strongly interacting, and
ultimately atomic, solid. In the specific model considered here, the dramatic increase of IR intensity
signals the onset of electron-vibron coupling in a
lower-symmetry environment. We have also pointed
out some novel aspects of CT in quantum lattices,
including the importance of the rotational quantum
number J, the absence of CT in phase-I p-hydrogen,
the modulation of t by zero-point motions, and the
pairwise CT for tilted molecules in an hcp lattice.
Open issues to be examined in future work include
CT contributions in phase II, vibronic differences
between deuterium and hydrogen, and intramolecular
versus intermolecular symmetry breaking.
Acknowledgements
We thank I.I. Mazin and R.E. Cohen for many
useful discussions. We gratefully acknowledge support of this work by the National Science Foundation
through DMR-9300163 (Princeton) and DMR9304028 (Geophysical Laboratory), and by NASA
through NAGW-1722 (Geophysical Laboratory). The
Center for High Pressure Research is an NSF Science and Technology Center.
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