Quantum biochemistry
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Edited by
Chérif F. Matta
Quantum Biochemistry
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Edited by
Che´rif F. Matta
Quantum Biochemistry
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Edited by
Chérif F. Matta
Quantum Biochemistry
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All books published by Wiley-VCH are carefully
produced. Nevertheless, authors, editors, and
publisher do not warrant the information contained
in these books, including this book, to be free of
errors. Readers are advised to keep in mind that
statements, data, illustrations, procedural details or
other items may inadvertently be inaccurate.
The Editor
Prof. Chérif F. Matta
Dept. of Chemistry & Physics
Mount Saint Vincent Univ.
Halifax, Nova Scotia
Canada B3M 2J6
and
Library of Congress Card No.: applied for
Dept. of Chemistry
Dalhousie University
Halifax, Nova Scotia,
Canada B3H 4J3
British Library Cataloguing-in-Publication Data
A catalogue record for this book is available from the
British Library.
Cover:
About the cover graphic (from Chapter 14):
A superimposition of (1) the electron density
r contour map of a Guanine-Cytosine WatsonCrick base pair in the molecular plane (the
outermost contour is the 0.001 e-/bohr3 isocontour followed by 2×10n, 4×10n, and 8×10n
e-/bohr3 with n starting at –3 and increasing in
steps of unity); and (2) representative lines of
the gradient of the density rr. The density is
partitioned into non-spherical color-coded “atomsin-molecules (AIM)”, each containing a single
nucleus. (Adapted from: C. F. Matta, PhD Thesis,
McMaster University, Hamilton, Canada, 2002).
(Courtesy of Chérif F. Matta).
Credit:
The phrase “Quantum Biochemistry” used in the title
of this book has been coined by Bernard Pullman and
Alberte Pullman (B. Pullman and A. Pullman,
Quantum Biochemistry; Interscience Publishers:
New York, 1963).
Bibliographic information published by
the Deutsche Nationalbibliothek
The Deutsche Nationalbibliothek lists this
publication in the Deutsche Nationalbibliografie;
detailed bibliographic data are available on the
Internet at .
# 2010 WILEY-VCH Verlag GmbH & Co. KGaA,
Weinheim
All rights reserved (including those of translation into
other languages). No part of this book may be
reproduced in any form – by photoprinting,
microfilm, or any other means – nor transmitted or
translated into a machine language without written
permission from the publishers. Registered names,
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specifically marked as such, are not to be considered
unprotected by law.
Printed in the Federal Republic of Germany
Printed on acid-free paper
ISBN: 978-3-527-32322-7
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To every experimentalist and theoretician who has contributed to Quantum Biochemistry,
and to every scientist, practitioner, and philosopher in whom its advancement, use, and
interpretation finds fruition.
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VII
Acknowledgment
This book is the result of the contributions of Ms. Alya A. Arabi, Dr. J. Samuel Arey,
Prof. Paul W. Ayers, Prof. Richard F.W. Bader, Dr. José Enrique Barquera-Lozada,
Dr. Joan Bertran, Dr. Michel Bitbol, Mr. Hugo J. Bohrquez, Prof. Russell J. Boyd,
Dr. Denis Bucher, Dr. Steven K. Burger, Prof. Roberto Cammi, Prof. Chiara Cappelli,
Dr. Constanza Cárdenas, Prof. Paolo Carloni, Dr. Lung Wa Chung, Dr. Fernando
Clemente, Prof. Fernando Cortés-Guzmán, Prof. Gabriel Cuevas, Prof. Matteo Dal
Peraro, Prof. Katherine V. Darvesh, Prof. Sultan Darvesh, Prof. Bijoy K. Dey,
Prof. Leif A. Eriksson, Dr. Laura Estévez, Dr. Michael J. Frisch, Prof. James W. Gauld,
Dr. Konstantinos Gkionis, Dr. María J. González Moa, Dr. Ana M. Gra, Dr. Anna V.
Gubskaya, Ms. Mireia Güell, Dr. Mark Hicks, Dr. J. Grant Hill, Dr. Lulu Huang,
Dr. Marek R. Janicki, Dr. Jerome Karle, Dr. Noureddin El-Bakali Kassimi, Prof. Eugene
S. Kryachko, Dr. Xin Li, Ms. Yuli Liu, Dr. Jorge Llano, Mr. Jean-Pierre Llored, Dr.
Marcos Mandado, Prof. Earl Martin, Prof. Lou Massa, Dr. Fanny Masson, Prof.
Robert S. McDonald, Prof. Benedetta Mennucci, Prof. Keiji Morokuma, Prof. Ricardo
A. Mosquera, Dr. Klefah A.K. Musa, Dr. Marc Noguera, Prof. Manuel E. Patarroyo,
Prof. Jason K. Pearson, Dr. James A. Platts, Prof. Paul L.A. Popelier, Prof. Ian R. Pottie,
Prof. Arvi Rauk, Dr. Arturo Robertazzi, Prof. Jorge H. Rodriguez, Dr. Luis RodríguezSantiago, Prof. Ursula Rưthlisberger, Ms. Debjani Roy, Ms. Lesley R. Rutledge, Dr.
Utpal Sarkar, Prof. Paul von Ragué Schleyer, Prof. Mariona Sodupe, Prof. Miquel Solà,
Dr. David N. Stamos, Dr. Marcel Swart, Prof. Ajit J. Thakkar, Prof. Jacopo Tomasi,
Prof. Alejandro J. Vila, Dr. Thom Vreven, Prof. Donald F. Weaver, Prof. Stacey D.
Wetmore, and Prof. Ada Yonath. I cannot thank each contributor enough for accepting
my invitation. I feel honored to have had the chance of working with such an
exceptional group of scientists.
The staff of Wiley-VCH has been instrumental in all phases of the development of
this project from its conception by copy-editing, proof reading, preparing galley
proofs, contacting authors, and for the timely production of this book. I have been
very lucky to work with them and extend my deepest thanks to Dr. Heike Noethe,
Dr. Eva-Stina Riihimäki, Dr. Ursula Schling-Brodersen, Dr. Martin Ottmar,
Ms. Claudia Nussbeck, and Ms. Hiba-tul-Habib Nayyer for their considerable effort,
professionalism, experience, and expertise on which I have constantly relied in the
past two years.
Quantum Biochemistry. Edited by Chérif F. Matta
Copyright Ó 2010 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim
ISBN: 978-3-527-32322-7
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VIII
Acknowledgment
I am very grateful to Prof. Lou Massa for his invaluable help in the form of opinion
and advice about the concept and design of this book.
I thank my colleagues and the administration at Mount Saint Vincent University,
past and present, for their moral and administrative support and continual encouragement. I am also indebted to Dalhousie University and the Université Henri Poincaré
(Nancy Université – 1) for access to their resources, including their libraries, by virtue
of, first, an ‘‘honorary Adjunct Professorship’’, and second, a ‘‘Visiting Professorship’’.
Extremely fortunate would be an understatement as to how I personally feel about
knowing, working with, and benefiting from the exceptional professional mentorship of Professors Richard F. W. Bader, Russell J. Boyd, Claude Lecomte, Lou Massa,
and John C. Polanyi. I cannot see how I could have edited this book without having
considerably benefited in numerous ways from my association with each.
The funding received by my research group was indispensable for the completion
of this project. I am much obliged to the Natural Sciences and Engineering Research
Council of Canada (NSERC), Canada Foundation for Innovation (CFI), and Mount
Saint Vincent University for financial support.
In closing, and on a more personal note, I wish to express my deepest and most
affectionate gratitude to the memory of those who gave me life: Farid A. Matta, and
Nabila Matta (née Nassif Abdel-Nour) for bringing me up in a rich and vibrant
intellectual atmosphere with a well-stocked library and art collection at our home in
Alexandria, and to the other members of our family who have always supported me
unconditionally, in particular during the unfolding of this demanding project:
Maged, Heba, Sara, and Nadine Matta.
Chérif F. Matta
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IX
Congratulations to Professor Ada Yonath for Winning the 2009
Nobel Prize in Chemistry
The editor this book and the staff of Wiley-VCH extend their warmest congratulations to Professor Ada Yonath for winning the 2009 Nobel Prize in Chemistry. They
undertake this opportunity to thank her again for her contribution to this book
(Chapter 16) that she has co-authored with Prof. Lou Massa, Prof. Chérif F. Matta,
and Dr. Jerome Karle.
Quantum Biochemistry. Edited by Chérif F. Matta
Copyright Ó 2010 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim
ISBN: 978-3-527-32322-7
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XI
Introductory Reflections on Quantum Biochemistry:
From Context to Contents
Cherif F. Matta
I will at least report novel properties of gases, the effects of which are regular, by proving that
these substances combine among each other in very simple ratios, and that the volume
contraction that they experience by the combination follows also a regular law. I hope to
provide through that a proof of what has been put forward by very distinguished chemists,
that we are perhaps not far from the epoch in which we will be able to submit to calculation
the majority of chemical phenomena.1)
Louis-Joseph Gay-Lussac, 31 December 1808 [1].
Two hundred and one years ago, almost to the day, Gay-Lussac (1778–1850) made the
far-reaching prediction that, one day, the majority of chemical phenomena will be
amenable to calculations. The boldness of this prediction is as extraordinary as the
accuracy with which it has been (and is being) realized. The history of science since
the early nineteenth century to the present is extremely rich and complex and studded
with important milestones that fall well beyond the scope of these short introductory
remarks and outside of the knowledge comfort zone of the writer, so only a few
relevant highlights will be offered to set the stage for this book. One of these
milestones was the award of the 1998 Nobel Prize in Chemistry, two centuries short
of a decade after Gay-Lussacs prediction, to Walter Kohn for his development of the
density-functional theory and to John Pople for his development of computational methods
in quantum chemistry. This visionary opening quotation, with wording such as
soumettre au calcul or submit to calculation, cannot have a more contemporary
ring!
1) Translated by the present writer from the
original text in French: «Je vais du moins faire
conntre des proprietes nouvelles dans les
gaz, dont les effets sont reguliers, en prouvant
que ces substances se combinent entre elles
dans des rapports tres-simples, et que la
contraction de volume quelles eprouvent par
la combinaison suit aussi une loi reguliere.
Jespere donner par la une preuve de ce quont
avance des chimistes tres – distingues, quon
nest peut-^etre pas eloigne de lepoque a laquelle on pourra soumettre au calcul la plupart
des phenomenes chimiques » [1]. (See
Figure 1).
Quantum Biochemistry. Edited by Chérif F. Matta
Copyright Ó 2010 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim
ISBN: 978-3-527-32322-7
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XII
Introductory Reflections on Quantum Biochemistry: From Context to Contents
Figure 1 The first two pages of L. J. Gay-Lussacs 1809 paper (Ref. [1]). The paper was read in the
last day of 1808 but was published in 1809. (The M. before the name of the author is the title
Monsieur, or Mr.)
The quotation is extracted from the second page of Gay-Lussacs 1809 paper [1] On
the combination of gaseous substances, one another (Figure 1). In this paper Gay-Lussac
applies the concepts of the modern atomic theory formulated by his contemporary,
John Dalton [2], to explain why gases combine in simple volumetric proportions. An
immediate progeny of Gay-Lussacs paper was one by Amedeo Avogadro
(1776–1856), who, in a single paper, introduced the concepts of mole, the number
later to be named in his honor NA, a method to calculate atomic and molecular
weights, and the distinction between elementary molecules [atoms] and
molecules [3]. Avogadros work led Stanislao Cannizzaro (1826–1910) to the
determination of atomic weights for the first time in 1858 [4]. Two years later, in
September 1860, Kekule, Wurtz, and Weltzien organized the Karlsruhe Congress
[5, 6], an international meeting that was attended by prominent chemists at the
time, later to evolve into the International Union of Pure and Applied Chemistry
(IUPAC) [5]. Among the participants in the 1860 meeting were the likes of
Cannizzaro but also less established young scientists including 26-year-old Dmitri
Ivanovich Mendeleev (and also 30-year-old Julius L. Meyer). Reprints of
Cannizzaros paper [4] were distributed to the participants [5], including Mendeleev
and Meyer, the principal characters in the following act in the historical drama of
chemistry culminating with the periodic classification of the elements, initially on
the basis of Cannizzaros atomic weights.
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Introductory Reflections on Quantum Biochemistry: From Context to Contents
In 1916, a century and eight years after Gay-Lussac read his Memoire before the
Societe de physique et de chimie de la Societe dArcueil, Gilbert Newton Lewis
(1875–1946) proposed his model of the chemical bond [7, 8]. Lewis recognized, for
the first time, the tendency of free atoms to complete the noble gas electronic shell
configuration and the central role played by the electron pair. Recognizing the
importance of electron pairing in 1916 [7, 8] before the advent of modern quantum
mechanics and the discovery of spin, is an extraordinary achievement. Without the
benefit of the knowledge of electronic spin, Lewis was compelled to go as far as
questioning the applicability of Coulombs law itself at very small distances:
Coulombs law of inverse squares must fail at small distances [7]. Lewiss paper has
marked, in the humble opinion of the writer, the conception of the modern electronic
theory of chemical bonding.
In 1929, at the dawn of the era of quantum mechanics, Paul A. M. Dirac
(1902–1984) opens his paper entitled Quantum Mechanics of Many-Electron System [9] by the, now well-known, statement:
The underlying physical laws necessary for the mathematical theory of a large part of
physics and the whole of chemistry are thus completely known, and the difficulty is only
that the exact application of these laws leads to equations much too complicated to be
soluble.
What Dirac meant is that the solution of Schr€
odinger equation, the wavefunction
Y, provides a complete description and thus contains all the information that can be
known about the system in a given quantum state. But since the Schr€odinger
equation can be solved exactly only for a very small number of very simple systems
(composed of one or two particles at the most), Dirac goes on to close the opening
paragraph to his paper wishing that [9]:
It therefore becomes desirable that approximate practical methods of applying quantum
mechanics should be developed, which can lead to an explanation of the main features of
complex atomic systems without too much computation.
Eighty years later, today in 2009, much of Diracs wish to develop approximate
methods to extend the application of quantum mechanics to complex atomic
systems has been realized, but the search for better and faster approximations to
solve the Schr€odinger equation remains a subject of prime importance and current
interest in theoretical and quantum chemical research. The need for these
approximate practical methods is particularly pertinent to quantum biochemistry
where quantum mechanics is applied to biological systems of staggering complexity,
unimaginable just a few decades ago.
The Born-Oppenheimer (BO) approximation, that electrons being much lighter
than nuclei are capable of readjusting their distribution instantaneously on the
time-scale of nuclear motion, is one of the most accurate and seminal approximations
in quantum chemistry. This approximation decouples the nuclear and electronic
Hamiltonians, a considerable simplification by virtue of which the nuclei move on a
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XIV
Introductory Reflections on Quantum Biochemistry: From Context to Contents
potential energy surface (PES) generated by solving the electronic Schr€odinger
equation for all possible nuclear geometries [10–12].2)
The concept of potential energy surface was advanced for the first time in 1931 by
Henry Eyring and Michael Polanyi in their treatment of the H ỵ H2 reaction [13].
The concept has been further developed by Polanyi and Eyring but also by F. W.
London, S. Sato, Philip M. Morse, and others [14–16]. Laidlers book [17] presents an
excellent exposition of the role of PES in chemical kinetics and dynamics as well as
biographies of 41 of the early pioneers in this field. The book edited by Back and
Laidler [16] is a compilation of commented reprints of a selection of key papers on
PES, dynamics, and kinetics including a reproduction of Savante Arrhenius 1889
paper on k ¼ AeÀEa =RT . An extraordinary collection of scholarly essays dedicated to
Michael Polanyi by leading scientists (including his son, John C. Polanyi, who went
on to win the 1986 Nobel Prize in Chemistry), economists, historians, and philosophers – a mix of disciplines that reflect the grandeur and the breadth of the intellect
of Michael Polanyi – was published in 1961 on the occasion of his 70th birthday [18].
Thus the BO approximation allows for a separate solution of the electronic and
nuclear problems. The solution of the electronic, time-independent, non-relativistic,
Born-Oppenheimer molecular Schr€
odinger equation represents much of modern
quantum chemistry (and quantum biochemistry), while the prediction of IR and
Raman spectra require the solution of the nuclear Schr€odinger equation.
Further approximations have given rise to the evolution of two equivalent branches
of electronic structure theory: Valence Bond (VB) theory and Molecular Orbital (MO)
theory. Valence bond theory was founded by W. H. Heitler and F. W. London, and
further developed by J. C. Slater, L. C. Pauling, E. A. Hylleraas and several others. The
theory is reviewed qualitatively in Paulings monograph The Nature of the Chemical
Bond [19] and in C. A. Coulsons Valence [20] and its updated version by R.
McWeenys Coulsons Valence [21]. VB theory has been reviewed in the recent books
by S. Shaik and P. Hiberty [22] and by G. A. Gallup [23].
F. Hund and R. S. Mulliken developed the Molecular Orbitals approach, to which
several others have also made substantial contributions, including J. Lennard-Jones,
J. C. Slater, E. H€
uckel, C. Coulson, and John Pople. A set of coupled differential
equations, one for each spin orbital, is obtained by the application of the variational
principle. The solution is obtained in the form of a single Slater determinant in an
iterative manner, the self-consistent field (SCF) approach, constituting what is now
known as the Hartree-Fock (H-F) method [12, 24–29].
The spherical symmetry of atoms enables a separation of variables that facilitates
the solution of the SCF problem. This advantage is lost in molecules, a problem that
was solved by the introduction of the linear combination of atomic orbitals (LCAO)
credited to Roothaan [30] and Hall [31]. The Roothan equations can be solved from
first principle (ab initio SCF theory) or through empirical parametrization and further
simplifying approximations (semi-empirical methods). Depending on how Coulom2) There are cases where the BO approximation breaks down. See for example Ref. [168]. These cases are
of considerable interest but of no implications in quantum biochemistry at the present stage of
knowledge, to the best of the writers knowledge.
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Introductory Reflections on Quantum Biochemistry: From Context to Contents
bic correlation is accounted for in post-Hartree-Fock methods, a hierarchy of
methods of different degrees of approximation is obtained.
An excellent commented exposition of reprints of early historical papers on MO
and VB theories is available in a recent book edited by H. Hettema [32]. The rivalry
between MO and VB theories has been the subject of a recent mind-stimulating
tripartite conversation between Roald Hoffman, Sason Shaik, and Philippe Hiberty, a
highly recommended reading [33].
A radically distinct approach to solve the electronic problem with the incorporation of Coulombic correlation, comparable in accuracy to post-HF methods
but with a considerable computational economy, is modern Density Functional
Theory (DFT) [34–36]. Perdew et al. [37] have recently published a very clear nonmathematical conceptual review of DFTs basic principles and ideas, an excellent
read.
While originally proposed by L. Tomas and E. Fermi, the modern formulation of DFT
was born in 1964 when P. Hohenberg and W. Kohn announced their celebrated (HK)
theorems [38]. The first HK theorem was reached through an elegant proof ad
absurdum that there exists a unique functional relationship between the external
potential and the electron density, and as a consequence, between the density and the
total energy of the system. The second theorem states that the exact electron density of
the ground state is one that minimizes the total energy. In other words, the second
theorem states that the variational principle can be invoked to calculate the energy of
the ground state. These powerful theorems in themselves offer no procedure to
compute the energy given the density. W. Kohn and L. Sham devised a workable
practical solution to this problem a year later, in 1965, when they cast the theory into a
formalism that resembles the Hatree-Fock SCF method in structure but with a
completely new meaning and interpretation of the (KS) orbitals [39]. The problem of
finding the exact functional remains unsolved to the time of writing.
DFT has evolved to become a formidable computational tool in the arsenal of the
solid state physicists, quantum and computational chemists, and computational
biochemists thanks to the subsequent pioneering work of Walter Kohn, Axel D.
Becke, Robert Parr, Weitao Yang, John Purdue, Donald Truhlar, Tom Ziegler and
others [34–36]. DFT has branched into a utilitarian/computational flavor used
extensively to generate the results similar to the ones reviewed in this book, but
also into a branch often called conceptual DFT aiming at deepening our understanding of the physical bases of chemistry and pioneered by the Belgium school
including P. Geerlings, F. De Proft, P. Bultinck, and the McMaster research group of
Paul Ayers, among others (see for example Refs. [40, 41].
The application of electronic structure calculations (wavefunction and density
functional methods) to real problems has been pushed to the forefront by scientists
such as John Pople, Paul von Rague Schleyer, Henry F. Schaeffer III, Leo Radom,
Warren J. Hehre, Keiji Morokuma, Jacopo Tomasi, Kendall Houk, and a number of
other pioneers [42–45]. A crowning achievement of the computational implementation of electronic structure methods is the development over several decades of very
sophisticated software such as GAUSSIAN [43, 46] and GAMESS [47] in molecular
quantum mechanics, and CRYSTAL [48, 49] in solid state physics.
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XVI
Introductory Reflections on Quantum Biochemistry: From Context to Contents
Electronic structure calculations, the primary focus of this book, represent a
principal branch of a wider field that can be called theoretical and computational
chemistry [44] and which includes, for example, molecular mechanics and force field
methods, Monte Carlo simulations, molecular dynamics simulations, molecular
modeling and docking, informatics, etc. [50–54].
The early uses of digital computers in chemistry marked the birth of computational
chemistry in the 1950s. This period coincided with spectacular advances in structural
biology that culminated in the discovery of the alpha-helical structure of DNA by
James Watson and Francis Crick [55–58] on the basis of a well-resolved X–ray
diffraction pattern obtained by Rosalind Franklin [59].
Interestingly, a book appeared in 1944 based on a series of lectures delivered a year
earlier at Trinity College, Dublin, in the midst of World War II (in 1943), by Erwin
Schr€
odinger. The book was not about wave mechanics but about biology viewed
through a physicists lens with the daring question What is life? as its title [60]. In this
book, the word code was used for the first time in the context of genetics when
Schr€
odinger described the chromosome as a code-script. In an incredibly unique
leap of insight, and in a section entitled The Variety of Contents Compressed in the
Miniature Code, Schr€
odinger writes [60]:
It has often been asked how this tiny speck of material, nucleus of the fertilized egg,
could contain an elaborate code-script involving all the future development of the
organism. A well-ordered association of atoms, endowed with sufficient resistivity to
keep its order permanently, appears to be the only conceivable material structure that
offers a variety of possible (isomeric) arrangements, sufficiently large to embody a
complicated system of determinations within a small spatial boundary. Indeed, the
number of atoms in such a structure need not be very large to produce an almost
unlimited number of possible arrangements. For illustration, think of the Morse
code. The two different signs of dot and dash in well-ordered groups of not more than
four allow thirty different specifications. Now, if you allowed yourself the use of a
third sign, in addition to dot and dash, and used groups of not more than ten, you
could form 88,572 different letters; with five signs and groups up to 25, the number
is 372,529,029,846,191,405.
That the gene is to be thought of as an information carrier, Watson says [58], was
the most important point made by Schr€odinger. Schr€odingers book was instrumental in its influence on a young generation of structural biologists that included
James Watson and Francis Crick. In fact, apparently it is What is Life? that ignited the
interest of Francis Crick to switch from physics to biology, as recounted by
Watson [58].
It is a particularly remarkable piece of history that Schr€odinger, the discoverer and
inventor of much of quantum mechanics, was also the one who planted many of the
seeds of modern structural and molecular biology, whether directly by underscoring
the importance of investigating the nature of information coding in the gene
(unknown at the time) or through his considerable influence on the careers,
enthusiasm, and thoughts of major players such as Watson and Crick. Thus the
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Introductory Reflections on Quantum Biochemistry: From Context to Contents
Figure 2 (a) Dust cover and (b) Abbreviated Table of Content of Quantum Biochemistry by
Bernard Pullman and Alberte Pullman published in 1963 [61]. Note how current the topics listed in
the table of content by todays standards, more than four decades after its publication.
phrase Quantum Biochemistry, coined in 1963 by Bernard and Alberte Pullman [61]
(Figure 2), while describing impeccably a definitive modern field of research whereby
quantum mechanics is applied to biological molecules and reactions, the subject of this book,
also epitomizes an era during which the synergy between physics and biology has benefited
humankind in a manner that is rarely encountered in human intellectual history.
The discovery of the chemical nature and structure of the genetic material has,
thus, brought biology within reach of the tools of a branch of applied quantum
mechanics, namely, quantum chemistry, which when applied to biological systems is
termed quantum biochemistry (QB). Among the earliest work in QB was the now
well-know mechanism of spontaneous and induced mutation, proposed by Per-Olov
L€
owdin in 1963, in which a mutation is the result of tautomeric transitions of the
two bases accompanied with double proton transfer by tunneling through the two
barriers of the pair of double potential wells, each corresponding to a hydrogen bond
linking the Watson-Crick partners [62, 63]. (See Chapter 31 of this book for a very
interesting review of this mechanism and its evolutionary consequences). If this
change in the hydrogen-bonding signature happens prior to transcription it results in
the incorporation of an erroneous base in mRNA and may lead to a non-silent
mutation if the altered codon is not a synonym of the original one. (An important
three-volume collective work dedicated to the memory of Per-Olov L€owdin has
recently been edited by E. J. Br€andas and E. S. Kryachko [64] and includes chapters
that review recent research done on this mechanism of mutation).
Another notable example of early insightful uses of computational quantum
chemistry in biology was the elucidation of the nature of the high energy phosphate
bond and the nature of its chelate with magnesium by Fukui et al. [65, 66]. Other early
efforts in QB were spearheaded by the Pullmans. They relied on early semiempirical
methods such as H€
uckel Theory or the PPP (Pariser-Parr-Pople) method to elucidate
the electronic structure of polycyclic aromatic hydrocarbons (PAHs) and correlate it
to carcinogenicity [67, 68], the electronic structure of nucleic acids [69], and to explore
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XVIII
Introductory Reflections on Quantum Biochemistry: From Context to Contents
stacking interactions between PAHs and nucleic acid bases [70]. Further examples
are reviewed in Pullman and Pullmans remarkable monograph Quantum
Biochemistry [61].
What is particularly commendable and admirable in the contribution of the
Pullmans is their boldness in attacking problems of biology by performing calculations on molecules of sizes reaching a few dozens of atoms at a time when the results
of ab initio calculations on diatomics were publishable in the best journals. To the
Pullmans credit also is their total mastery of both the biology and the physics and
their ability to look beyond the calculation to the larger picture, evolutionary biology
being a noted example [71]. A glance at the table of content of their book cannot
convey a more timely impression even today in 2009 (Figure 2). This present book
aims at contributing to review the state-of-the art of quantum biochemistry supplementing several excellent other books that have a similar goal (see for example
Refs. [72–76]).
Naturally, the transformation of theoretical chemistry into computational chemistry has been greatly facilitated not only by the very fast increase in the power and
availability of computers but also by the development of methods tailored for large
molecules as they occur in quantum biochemistry. In the 1960s, performing an ab
initio calculation on a small molecule composed of a handful of atoms represented the
limit of what could be achieved. Nowadays, computational strategies have allowed for
the calculation of ever increasingly large and complex systems.
In recent years, the need to study enzyme active sites under the influence of
the surrounding (whether the surroundings are the remainder of the protein, of the
immediate surrounding amino acid residues near the active site) has provided the
impetus for the development of methods that treat the active site of interest at
the highest achievable computational level of theory and treating the surrounding
as the source of a perturbing field at lower (more economical) level(s) of theory,
hence optimizing the balance of accuracy and speed. If the active site is treated
quantum mechanically (QM) and the remainder of the protein by molecular
mechanics (MM) the method is known as QM/MM [77, 78]. Hybrid methods
have found numerous applications in biochemistry and are now a standard and
very powerful tool in the hands of quantum and computational biochemists.
(See Chapters 2, 3, 4, and 17 of this book for excellent reviews on hybrid quantum
mechanical methods).
Another important breakthrough concerned with very large systems such as
proteins and nucleic acids is the reconstruction of the density matrix of the target
macromolecule from density matrices of its composing pieces termed kernels.
This method, developed in its present form by Lulu Huang, Lou Massa, and Jerome
Karle, the subject of the opening chapter of this book, is termed Quantum
Crystallography (QCr) and is also sometimes referred to as the Kernel Energy
Method(KEM).
The QCr/KEM method has been rigorously and repeatedly tested by comparing ab
initio wavefuctions obtained directly on full molecules to the corresponding wavefunctions reconstructed from kernels. This repeated benchmarking has established
the accuracy and validity of this approximation. The crowning achievement of this
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Introductory Reflections on Quantum Biochemistry: From Context to Contents
Figure 3 The crystal structure of vesicular
stomatitis virus nucleocapsid protein Ser290Trp
mutant (2QVJ) [87] (a) ribbon model (b) atomic
model (without hydrogen atoms). The ab initio
energy of this gigantic molecule composed of
some 33,175 has been calculated using the
Kernel Energy Method [79]. This is the largest ab
initio calculation known to the writer at the time
of writing.
approach has been the calculation of the Hartree-Fock [HF/6-31G(d,p)] energy as well
as the MP2/6-31G(d,p) interaction energies within the vesicular stomatitis virus
nucleoprotein, a protein composed of a staggering 33,175 atoms (Figure 3) [79]. This
result has been the fruit of decades of development going back to the late 1960s
[80–82] and more recently with applications to very large molecules such as DNA [83],
tRNA [84], the ribosome [85], and insulin [86].
Solvation is another area of prime importance to the quantum chemistry of biological
molecules. While solvation is still not considered as a solved problem in quantum
chemistry, considerable advances have been achieved already. Solvent effects are
commonly accounted for by either (a) the explicit incorporation of solvent molecules
into the quantum mechanical calculation, sometimes referred to as the supermolecule approach, or (b) implicit solvation known as the self-consistent reaction
field (SCRF) approach in which the solute is placed in a cavity inside the solvent (the
shape of this cavity depends on the particular model chosen). The solvent is then
modeled as a continuum characterized by its uniform dielectric constant [88–91]
Scientists such as Jacopo Tomasi, Donald Truhlar, and Cristopher Cramer are among
the pioneers in this field. (See Chapter 4 for an authoritative review).
The discovery of solutions to the phase problem of X-ray crystallography, e.g., the
discovery of direct methods by Jerome Karle and Herbert A. Hauptman (the Nobel
Laureates in Chemistry for 1985), the dramatic engineering advances in the design of
diffractometers and of data collection devices, most notably, the invention of the CCD
(charge-coupled device) camera, and the advent of bright synchrotron X-ray sources,
all contributed to an unprecedented shortening of the data collection and structure
solution times. As a result, the solution of X-ray crystallographic structures has
become standardized and faster than ever. Incidentally, the invention of the CCD is a
theme of the 2009 Nobel Prize in Physics awarded to Willard S. Boyle and George E.
Smith.
As a result of these exciting developments, and because of the widespread
availability of the internet, we are now witnessing an exponential proliferation of
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massive databases of structural information. Besides the deposition of crystallographic information files (cif) as electronic supplementary material to published
articles, there are now several repositories of structural information, and to name a
few important examples we list The Cambridge Structural Database (CSD), the
Crystallography Open Database (COD), the Nucleic Acid Database, and the Protein
Data Bank (PDB).
The largest object that has been crystallized to this day is the ribosome, a task
generally believed impossible just a few years ago. The crystalization of the ribosome
and the solution of its structure are achievements of epical proportions because they
provide the atomic details necessary to understand how it reads the genetic information encoded in the mRNA and how it translates this information into a
polypeptide. This is tantamount to uncovering one of lifes most jealously guarded
secrets. The implications of this fundamental knowledge are considerable for
example in the design of selective protein synthesis inhibitors, i.e., antibiotics that
selectively target the ribosomes of harmful bacteria leaving human ribosomes intact.
Venkatraman Ramakrishnan, Thomas A. Steitz, and Ada E. Yonath were awarded the
2009 Nobel Prize in Chemistry for solving the difficult jigsaw puzzle leading to the
full atomic structure of the ribosome. Besides her contributions in working out key
aspects of the structure and function of the ribosome, Ada Yonath is also credited for
the development of an entirely new technique termed cryo-bio-crystallography,
indispensable for the crystallization and subsequent solution of the ribosomal
architecture [92]. Ada Yonath is the fourth women to win the Prize in Chemistry,
joining the league of Marie Curie (1911), Irene Joliot-Curie (1935), and Dorothy
Crowfoot Hodgkin (1964).
Besides its primary role in yielding structural information about molecules of
widely varying sizes and chemical composition, X-ray crystallography has also
evolved into another direction concerned with the nature of the chemical bond
in Paulings words. In a routine crystallographic data treatment, the experimental
structure factors are refined by iterative comparison with those obtained by a reverse
Fourier transform of a model density. The model density of the unit cell is obtained
from a guessed structure where spherical atomic densities are placed at the positions
of the nuclei assumed in the model [93]. Only the atomic positions are allowed to
change during the refinement cycles but not their spherical shape. This approach is
suitable for molecular geometries but is not capable of capturing the subtle deformations of the electron density in regions relatively removed from the nuclei, as in the
regions of chemical bonding. For that purpose, an aspherical multipolar refinement
strategy is necessary [94]; a widely used multipolar model is that of Hansen and
Coppens [95–97].
When the quality of a crystal is good and if the experiment is carefully conducted
(preferably at very low temperatures) followed by the appropriate corrections and
multipolar refinement, it can yield very accurate electron density maps of the
bonding regions. The question now is how to analyze these electron density maps? How
to extract the chemistry folded and encoded within the density? These questions are
equally valid with reference to the output of the electronic structure calculations
described above.
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The answers to these important questions are rooted in the early 1960s, when
Richard F. W. Bader et al. calculated and analyzed ab initio molecular electron density
distributions well before the electron density was an object of intense interest
[98–100]. In 1963 Richard F. W. Bader and Glenys A. Jones write [99]:
The manner in which the electron density is disposed in a molecule has not
received the attention its importance would seem to merit. Unlike the energy of a
molecular system which requires a knowledge of the second-order density matrix
for its evaluation [101] many of the observable properties of a molecule are
determined in whole or in part by the simple three-dimensional electron-density
distribution. In fact, these properties provide a direct measure of a wide spectrum
of different moments averaged directly over the density distribution. Thus the
diamagnetic susceptibility, the dipole moment, the diamagnetic contribution to the
nuclear screening constant, the electric field, and the electric field gradient (as
obtained from nuclear quadrupole
coupling
constants)
provide
a measure of (aside
from any angular dependencies) ri2 , hri i, riÀ1 , riÀ2 , and riÀ3 , respectively. The
electric field at a nucleus due to the electron density distribution is of particular
interest due to the theorem derived by Hellmann [102] and Feynman [103]. They
have demonstrated that the force acting on a nucleus in a molecule is determined by
the electric field at that nucleus due to the other nuclei and to the electron-density
distribution.
Over the past three decades, Bader and his students have constructed a theory of
great elegance, beauty, generality, and power. This theory is referred to in the older
literature as the Theory of Atoms-in-Molecules (AIM), and in the more recent
literature as the Quantum Theory of Atoms in Molecules(QTAIM) [104–109]. The
theory in one stroke provides a framework to discuss, classify, and understand
chemical structure and its (in)stability and transformations, chemical bonding
interactions (note the usage as a verb [110]), and a coherent and physically and
mathematically sound partitioning of the molecular space into individual atoms,
hence the designation Atoms-in-Molecules. The partitioning of the molecular
space into non-overlapping non-spherical atoms (see the cover graphic of this book)
allows the partitioning of any molecular property that can be expressed as a local
density into additive atomic and group contributions. In doing so, the theory has been
shown on numerous occasions to recover experimental transferability and additivity
schemes [104].
The theory has deep roots in quantum mechanics [111] and is founded on the
analysis of Dirac observables (see Chapter 14 of this book for a brief introduction).
The theory presents an interpretative and predictive scheme for chemistry that
parallels experiment (see Refs. [112, 113]).
It has recently been proposed to re-name QTAIM as Quantum Chemical
Topology and detailed and very compelling arguments to do so have been
presented [114]. However, in the present writers view, changing the designation
that everyone uses The Quantum Theory of Atoms in Molecules to another
designation is not recommended because it can cause confusion in the vast
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literature on the subject and will complicate literature searches. As a result, this is
likely to diminish the impact of the theory. More important, perhaps, is that
changing the designation of the theory may lead to the dilution of the credit that its
principal developer, Richard F. W. Bader, deserves. Finally, in the opinion of this
writer, it is incumbent on the principal developer of the theory to choose how to
name it. Ref. [114] is a highly recommended reading.
QTAIM is becoming the standard theory used to interpret and analyze experimental charge densities [96, 97, 115–122] and has gained a broad acceptance in the
computational chemistry community (as several of the chapters of this book show).
QTAIM has been extensively applied to calculated and experimental electron densities [96] to predict and interpret molecular properties at an atomic resolution,
including for example, heats of formation [123], magnetic susceptibilities [124, 125],
atomic electrostatic moments and polarizabilities [126, 127] Raman intensities [126–
129], IR intensities [130, 131], electron localization and delocalization [132, 133],
pKa [134], biological and physicochemical properties of the amino acids [135], protein
retention times [136], HPLC column capacity factors [137], and NMR spin-spin
coupling constants [138, 139]. The theory was also applied in the design of protein
force fields by atom typing [140], to automate the search for pharmacophores and/or
(re)active sites in a series of related molecules [141–145] and to reconstruct large
molecules not amenable to direct computation [146–148] or easy crystallization [120]
from transferable fragments.
In most of these studies, the analysis is applied to stationary points on the PES and,
generally, in the absence of external perturbations such as external fields (with the
exception of studies of polarizabilities). The advent of time-resolved crystallography,
pioneered by scientists such as Philip Coppens, has brought the fourth dimension
into the world of the experimental electron density [149–151]. A pump-probe
approach is used to first excite the crystal with ultra-short laser or X-ray pulses
followed by the interrogating pulse(s), the latter often polychromatic (Laue technique) to improve the time resolution. The work has generated images of the
electron density and its deformation upon electronic excitation and allowed a realtime observation of the change in the geometry of molecules upon charge transfer
induced by the external perturbation. Experimental activation energies have been
measured by analyzing the temperature-dependence of the rate constant of photoisomeration [152]. Paralleling these exciting experimental advances on the theoretical
side, studies that analyze the topology of the electron density as it evolved over the full
PES landscape, or along the steepest path of descent from TS to the reactants and
products valleys, the so-called reaction path (RP) [153–156], started to appear in the
literature [157–160].
Further, there exists a bijective mapping between the points of a PES and the
corresponding points belonging to each property surface such as dipole moment or
polarizability surfaces [161, 162]. Examples of such surfaces for the reaction F. ỵ
CH4 ! HF ỵ . CH3, are displayed in Figure 4.
In the presence of an external laser field, and at the low frequency limit, the
effective potential along the reaction path (X ỵ CH4, C3v symmetry) can be approximated by [161]: V ¼ VðsÞÀmðsÞeo cosðwÞ À 12 azz ðsÞe2o cos2 ðwÞ, where V(s) is the laser-
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