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arXiv:physics/0412180v2 [physics.chem-ph] 8 Apr 2005

On Emerging Field of Quantum

Chemistry at Finite Temperature

Liqiang Wei

Institute for Theoretical Atomic, Molecular and Optical Physics

Harvard University, Cambridge, MA 02138

February 2, 2008

Abstract

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.

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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.

1 Introduction

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 [1]. 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

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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

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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 [63]. 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.

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2Experimental 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.

2.1Temperature 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 [9]. The resulting spectra for the

absorption at T = 80 and 300K are shown in Figure 1 of that paper. We

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