arXiv:physics/0412180v2 [physics.chem-ph] 8 Apr 2005
On Emerging Field of Quantum
Chemistry at Finite Temperature
Institute for Theoretical Atomic, Molecular and Optical Physics
Harvard University, Cambridge, MA 02138
February 2, 2008
In this article, we present an emerging field of quantum chemistry
at finite temperature. We discuss its recent developments on both
theoretical and experimental fronts. We describe and analyze several
experimental investigations related to the temperature effects on the
structure, electronic spectra, or bond rupture forces for molecules.
This includes the study of the temperature impact on the pathway
shifts for the protein unfolding by atomic force microscopy (AFM),
the temperature dependence of the absorption spectra of electrons
in solvents, and temperature influence over the intermolecular forces
measured by the AFM. On the theoretical side, we review a recent
advancement made by the author in the coming fields of quantum
chemistry at finite temperature. Starting from Bloch equation, we
have derived the sets of hierarchy equations for the reduced density
operators in both canonical and grand canonical ensembles. They
provide a law according to which the reduced density operators vary
in temperature for the identical and interacting many-body particles.
By taking the independent particle approximation, we have solved the
equation in the case of a grand canonical ensemble, and obtained an
eigenequation for the molecular orbitals at finite temperature. The
explicit expression for the temperature-dependent Fock operator is
also given. They will form a foundation for the study of the molecular
electronic structures and their interplay with the finite temperature.
Furthermore, we clarify the physics concerning the temperature ef-
fect on the electronic structure or processes of the molecules which is
crucial for both theoretical understanding and computational study.
Finally, we summarize our discussion and point out the theoretical
and computational issues for the future explorations in the fields of
quantum chemistry at finite temperature.
Keywords Quantum chemistry at finite temperature; temperature depen-
dent; polymers; protein folding; intermolecular forces; solved electrons.
The history for quantum chemistry development is almost synchronous to
that of quantum mechanics itself. It begins with Heitler and London’s study
of electronic structure of H2molecule shortly after the establishment of wave
mechanics for quantum particles . There are two major types of molecu-
lar electronic theories: valence bond approach vs. molecular orbital method
with the latter being the popular one for the present investigation. It has
gone through the stages from the evaluation of molecular integrals via a
semiempirical way to the one by an ab initio method. Correlation issue is
always a bottleneck for the computational quantum chemistry and is un-
der intensive study for over fifty years. For large molecular systems such
as biomolecules and molecular materials, the development of the combined
QM/MM approach, pseudopotential method and linear scaling algorithm
has significantly advanced our understanding of their structure and dynam-
ics. There are about eight Nobel prize laureates whose researches are related
to the molecular electronic structure theory. This not only recognizes the
most eminent scientists who have made the outstanding contributions to the
fields of quantum chemistry, but more importantly, it indicates the essen-
tial roles the electronic structure theory has been playing in the theoretical
chemistry as well as for the whole areas of molecular sciences. Nowadays,
quantum chemistry has been becoming a maturing science [2, 3].
Nevertheless, the current fields of quantum chemistry are only part of the
story for the molecular electronic structure theory. From the pedagogical
points of view, the quantum mechanics based on which the traditional quan-
tum chemistry is built is a special case of more general quantum statistical
mechanics [4, 5, 6]. In reality, the experimental observations are performed
under the conditions with thermodynamic constraints. Henceforth, there is
a need to extend the current areas of quantum chemistry to the realm of, for
instance, finite temperature [4, 5, 6].
Indeed, many experimental investigations for various fields and for dif-
ferent systems have already shown the temperature or pressure effects on
their microscopic structures [7-30,49-57,63,71-87]. The polymeric molecules
are one of the most interesting systems for this sort of studies [7-16]. The ex-
perimental measurement on the absorption spectra, photoluminescence (PL),
and photoluminescence excitation (PLE), and spectral line narrowing (SLN)
for the PPV and its derivatives all show the same trend of the blue shift
with the increasing temperature [7, 8, 9]. This attributes to the temperature
dependence of their very rich intrinsic structures such as the vibronic cou-
pling [14, 15, 16]. The experimental investigation of the temperature effect
on the biomolecules started in the late nineteenth century [17, 18]. Most re-
cently, it has been extended to the study of folding and unfolding of protein
or DNA [19, 20, 21]. In addition to the observed patterns for the unfolding
forces with respect to the extension or temperature, it has been proved that
the temperature-induced unfolding is another way for the study of mech-
anisms or pathways of protein folding or unfolding processes [19-23]. The
newest related development is on the AFM measurement made by Lo et al.
of the intermolecular forces for the biotin-avidin system in the temperature
range from 286 to 310K . It has shown that an increase of temperature
will almost linearly decrease the strength of the bond rupture force for the
individual biotin-avidin pair. The study of temperature effect on the absorp-
tion spectra of solvated electron began in the 1950’s and it is still of current
interest. A striking effect is that an increasing temperature will cause the
positions of their maximal absorption red shift [71-85].
In recent papers, we have deduced an eigenequation for the molecular
orbitals [4, 5]. It is the extension from the usual Hartree − Fock equation at
zero temperature to the one at any finite temperature [88, 89]. It opens an
avenue for the study of the temperature effects on the electronic structures
as well as their interplay with the thermodynamic properties. In the third
section, we will present this equation and give the details for its derivation.
In the next section, we will show four major types of experiments related
to the study of the temperature influences over the microscopic structure
of molecular systems. In the final section, we will discuss and analyze our
presentations, and point out both theoretical and computational issues for
the future investigation.
2 Experimental Development
In this section, we mainly describe the experimental investigations related
to the temperature effect on the bonding, structure and electronic spectra of
molecules. We choose four kinds of the most recent developments in these
fields which are of chemical or biomolecular interests.
2.1 Temperature effects on geometric structure and
UV-visible electronic spectra of polymers
The first important systems where the important issues related to the tem-
perature effect on the geometric structure and electronic spectra are the
polymeric molecules. Many experimental investigations and some theoreti-
cal work already exist in the literature [7-16]. However, how the temperature
changes the microscopic structures of the polymers are still not completely
understood and there are many unresolved issues in interpreting their elec-
tronic spectra. We list here a few very interesting experimental investigations
for the purpose of demonstration.
The poly(p-phenylenevinylene)(PPV ) is one of the prototype polymeric
systems for the study of their various mechanical, electronic, and optical
properties. The impact from the temperature on the absorption spectra,
photoluminescence (PL), and photoluminescence excitation (PLE) of the
PPV have also been investigated both experimentally and theoretically [7,
8, 9]. In an experiment carried out by Yu et al., the absorption spectra are
measured for the PPV sample from the temperature 10 to 330K. The details
of the experiment are given in their paper . The resulting spectra for the
absorption at T = 80 and 300K are shown in Figure 1 of that paper. We
see that there is a pronounced change in the spectra when increasing the
temperature. They also study the PL and PLE spectra for the PPV. The
measured PL spectra at two temperatures: 77 and 300K are demonstrated
in the Figure 3, and the PLE spectra at those temperatures are depicted in
the Figure 4 of the paper . They both show the dramatic changes of the
band blue shift when the temperature is increased. Similar studies have also
been performed before by the other groups [7, 8]. They observed the similar
Another interesting investigation is related to the temperature effect on
the spectral line narrowing (SLN) of the poly(2-methoxy-5-(2
1,4-phenylenevinylene)(MEH − PPV) spin-coated from either THF or CB
solvents . In the experiment done by Sheridan et al., the SLN is mea-
sured together with the absorption and PL as shown in Fig. 1 of their paper.
It is found that the same trend of the SLN blue shift is observed as that for
the absorption and PL with an increasing temperature. They attribute this
to the same reason of the electronic structure modification resulting from the
variation of the temperature.
2.2 Temperature effects on structure, dynamics, and
folding/unfolding of biomolecules
Biomolecules are complex systems, featuring a large molecular size, a hetero-
geneity of atomic constitutes and a variety of conformations or configurations.
Their energy landscape thereby exhibits multiple substates and multiple en-
ergy barrier, and varies in size for the barrier heights [24, 25, 26, 27, 28]. The
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