Matta, Boyd. The quantum theory of atoms in molecules

XXX List of Abbreviations Appearing in this Volume

NBO	Natural bond orbital(s) (method)
NCP or ncp	Nuclear critical point
NICS	(Schleyer’s) nucleus-independent chemical shift
NICS(0)	(Schleyer’s) nucleus-independent chemical shift in the plane of an
	aromatic ring at its geometrical center
NICS(n)	˚
	(Schleyer’s) nucleus-independent chemical shift n A above the
	plane of an aromatic ring at its geometrical center
NMR	Nuclear magnetic resonance (spectroscopy)
NNA	Non-nuclear attractor
NNA	Nearest-neighbor approximation
NNM	Non-nuclear maximum (or maxima)
NPA	Natural population analysis
NR	Newton–Raphson (method/technique)
ODM	One-electron density matrix
OG	(Gas-phase) optimized geometry (as opposed to crystal geometry)
PAHB	Polarization-assisted hydrogen bond(ing)
PAH(s)	Polycyclic aromatic hydrocarbon(s)
PASA	Promolecular atom shell approximation
PBH(s)	Polybenzenoid hydrocarbon(s)
PDI	para-Delocalization index (of Sola` et al.)
PE	Potential energy
PES	Potential energy surface(s)
PEST	Property-encoded surface translator
PH	Poincare´-Hopf relationship
ppm	Part per million
Pyr	Pyridine
QCISD	Quadratic configuration interaction with single and double excita-
	tions
QCISD(T)	Quadratic configuration interaction with single and double excita-
	tions (with correction for triple excitations)
QCT	Quantum chemical topology
QSAR	Quantitative structure–activity relationship(s)
QSPR	Quantitative structure–property relationship(s)
QTAIM	Quantum theory of atoms in molecules
QTAMC	Quantum theory of atoms in molecules and crystals
RAHB	Resonance-assisted hydrogen bond(ing)
RCP or rcp	Ring critical point
RECON	(Rapid TAE) reconstruction (of molecular charge densities)
RHF	Restricted Hartree–Fock (method)
RNA	Ribonucleic acid
RSS	Root summed squares
R.H.S.	Right-hand side (of an equation)
SCF	Self-consistent field
SCVS	Self-consistent virial scaling
SES	Sodium electrosodalite

	List of Abbreviations Appearing in this Volume	XXXI
SF	Source function
SI	Surface integral
SIEP	Surface integral of the electrostatic potential
SIG	Surface integral of the G kinetic energy density
SIK	Surface integral of the K kinetic energy density
SSHB	Short strong hydrogen bond(ing)
std	Standard deviation
STM	Scanning–tunneling microscope (or microscopy)

TThymine

TAE	(Breneman’s) transferable atom equivalent
TMS	Transition metal sulfide(s)
TMS	Tetramethylsilane

UUracil

UHF	Unrestricted Hartree–Fock (method)
VHTS	Virtual high-throughput screening
VSCC	Valence shell charge concentration
VSCD	Valence shell charge depletion
VSEPR	Valence-shell electron-pair repulsion model of molecular geometry
XRD	X-Ray di raction
ZFS	Zero-flux surface

XXXIII

List of Contributors

Dr. Ibon Alkorta	Dr. Miguel A. Blanco
Instituto de Quı´mica Me´dica	Departamento de Quı´mica Fı´sica
(CSIC)	y Analı´tica
Juan de la Cierva 3	Universidad de Oviedo
28006 Madrid	33006 Oviedo
Spain	Spain
[e-mail]	[e-mail]
Dr. Yosslen Aray	Prof. Russell J. Boyd
Centro de Quimica, IVIC	Department of Chemistry
Apartado 21827	Dalhousie University
Caracas 1020 A	Halifax, NS, B3H 4J3
Venezuela	Canada
[e-mail]	[e-mail]
Prof. Richard F.W. Bader	Prof. Curt M. Breneman
Department of Chemistry	Department of Chemistry and
McMaster University	Chemical Biology
1280 Main Street West	Rensselaer Polytechnic Institute
Hamilton, ON, L4S 4M1	110 8th St
Canada	Troy, NY 12180-3590
[e-mail]	USA
	[e-mail]
Prof. Axel D. Becke
Department of Chemistry	Prof. Fernando Corte´s-Guzma´n
Dalhousie University	Departamento de Quı´mica Orga´nica
Halifax, NS, B3H 4J3	Facultad de Quı´mica
Canada	Universidad Nacional Auto´noma de
[e-mail]	Me´xico
	Circuito Escolar, Ciudad Universitaria
	Coyoaca´n, Me´xico D.F. 04510
	[e-mail]

XXXIV	List of Contributors
	Dr. Aurora Costales	Prof. Jose´ Elguero
	Departamento de Quı´mica Fı´sica	Instituto de Quı´mica Me´dica (CSIC)
	y Analı´tica	Juan de la Cierva 3
	Universidad de Oviedo	28006 Madrid
	33006 Oviedo	Spain
	Spain	[e-mail]
	[e-mail]
		Prof. Enrique Espinosa
	Prof. Gabriel Cuevas	Laboratoire de Cristallographie et
	Departamento de Fisicoquı´mica	Mode´lisation des Mate´riaux Mine´raux
	Instituto de Quı´mica	et Biologiques
	Universidad Nacional Auto´noma	Faculte´ des Sciences et Techniques
	de Me´xico	Universite´ Henri Poincare´
	Circuito Escolar, Ciudad	Nancy 1
	Universitaria	BP 239, 54506 Vandoeuvre-le´s-
	Coyoaca´n, Me´xico D.F. 04510	Nancy, Cedex
	Me´xico	France
	[e-mail]	[e-mail]
	Dr. Richard Dawes	Dr. Carlo Gatti
	Department of Chemistry	CNR-ISTM Istituto di Scienze e
	University of Missouri-Columbia	Tecnologie Molecolari
	Columbia, MO 65211	Via Golgi 19
	USA	20133 Milano
	[e-mail]	Italy
		[e-mail]
	Dr. Birger Dittrich
	School of Biomedical & Chemical	Prof. Ronald J. Gillespie
	Sciences	Department of Chemistry
	The University of Western	McMaster University
	Australia	1280 Main Street West
	35 Stirling Highway	Hamilton, ON, L4S 4M1
	Crawley WA 6009	Canada
	Australia	[e-mail]
	[e-mail]	[e-mail]
	[e-mail]
		Prof. Kathleen M. Gough
	Dr. Jason R. Dwyer	Department of Chemistry
	Max-Born-Institut fu¨r	University of Manitoba
	Nichtlineare Optik und	Winnipeg, MB, R3T 2N2
	Kurzzeitspektroskopie	Canada
	Max-Born-Strasse 2A	[e-mail]
	12489 Berlin
	Germany
	[e-mail]

List of Contributors XXXV

Prof. Sławomir J. Grabowski

Department of Physics and Chemistry

University of Ło´dz´ ul.Pomorska 149/153 90-236 Ło´dz´

Poland [e-mail]

Dr. Benoıˆt Guillot

Laboratoire de Cristallographie et Mode´lisation des Mate´riaux Mine´raux et Biologiques

LCM3B (UMR UHP-CNRS 7036) Faculte´ des Sciences et Techniques

Universite´ Henri Poincare´ BP 239, Boulevard des Aiguillettes

54506 Vandoeuvre-le`s-Nancy CEDEX

France [e-mail]

Dr. Christopher E. Henze

MS T27-A

NASA Advanced Supercomputing Division

NASA Ames Research Center Mo ett Field, CA 94035 USA

[e-mail]

Prof. Jesu´s Herna´ndez-Trujillo

Departamento de Fı´sica y Quı´mica Teo´rica Facultad de Quı´mica

Universidad Nacional Auto´noma de Me´xico

Circuito Escolar, Ciudad Universitaria Coyoaca´n, Me´xico D.F. Me´xico 04510

[e-mail]

Dr. Christian Jelsch

Laboratoire de Cristallographie et Modelisation des Mate´riaux Mine´raux et Biologiques Unive´rsite´ Henri Poincare´ LCM3B (UMR UHP-CNRS 7036) Faculte´ des Sciences et Techniques Universite´ Henri Poincare´

BP 239, Boulevard des Aiguillettes 54506 Vandoeuvre-le`s-Nancy CEDEX France [e-mail]

Dr. Todd A. Keith

Department of Chemistry University of Missouri – Kansas City 5100 Rockhill Road

Kansas City, MO 64110 USA [e-mail]

Dr. Ange´lique Lagoutte

Laboratoire de Cristallographie et Modelisation des Mate´riaux Mine´raux et Biologiques Unive´rsite´ Henri Poincare´ LCM3B (UMR UHP-CNRS 7036) Faculte´ des Sciences et Techniques Universite´ Henri Poincare´

BP 239, Boulevard des Aiguillettes 54506 Vandoeuvre-le`s-Nancy CEDEX France [e-mail]

Prof. Claude Lecomte

Laboratoire de Cristallographie et Modelisation des Mate´riaux Mine´raux et Biologiques Unive´rsite´ Henri Poincare´ LCM3B (UMR UHP-CNRS 7036) Faculte´ des Sciences et Techniques Universite´ Henri Poincare´

BP 239, Boulevard des Aiguillettes 54506 Vandoeuvre-le`s-Nancy CEDEX France [e-mail]

XXXVI List of Contributors

Prof. Laurence Leherte

Laboratoire de Physico-Clinic Informatique (PCI)

Faculte´s Universitaires NotreDame de la Paix

Rue de Bruxelles 61 5000 Namur Belgium

[e-mail]

Prof. Vı´ctor Luan˜a

Departamento de Quı´mica Fı´sica y Analı´tica

Universidad de Oviedo 33006 Oviedo

Spain [e-mail]

Prof. Peter Luger

Institute for Chemistry/ Crystallography

Free University of Berlin Takustr. 6

14195 Berlin Germany

[e-mail]

Dr. Piero Macchi

Dipartimento di Chimica Strutturale e Stereochimica Inorganica

Universita` di Milano Via Venezian 21 20133 Milano

Italy

and

CNR – Istituto di Scienze e Tecnologie Molecolari

Via Golgi 19 Milano Italy

[e-mail]

Prof. Preston J. MacDougall

Department of Chemistry

Middle Tennessee State University 1301 E. Main St.

Murfreesboro, TN 37132 USA [e-mail]

[e-mail]

Dr. Ignasi Mata

Institut de Cie`ncia de Materials de Barcelona (CSIC)

Campus UAB

08193 Bellaterra Spain [e-mail]

Dr. Eduard Matito

Institut de Quı´mica Computacional Departament de Quı´mica Universitat de Girona

Campus de Montilivi 17071 Girona Catalonia

Spain [e-mail]

Prof. Che´rif F. Matta

Department of Chemistry & Physics

Mount Saint Vincent University

Halifax, NS, B3M 2J6

Canada

and

Department of Chemistry Dalhousie University Halifax, NS, B3H 4J3 Canada [e-mail] [e-mail]

	List of Contributors	XXXVII
Prof. Elies Molins	Dr. Jordi Poater
Institut de Cie`ncia de Materials	Afdeling Theoretische Chemie
de Barcelona (CSIC)	Scheikundig Laboratorium der Vrije
Campus UAB	Universiteit
08193 Bellaterra	De Boelelaan 1083
Spain	NL-1081 HV Amsterdam
[e-mail]	The Netherlands
	[e-mail]
Dr. Paula Mori-Sa´nchez
Departamento de Quı´mica Fı´sica	Dr. Paul L. A. Popelier
y Analı´tica	School of Chemistry
Universidad de Oviedo	University of Manchester
33006 Oviedo	Faraday Building, North Campus
Spain	Manchester M60 1QD
[e-mail]	UK
	[e-mail]
Prof. Angel Martı´n Penda´s
Departamento de Quı´mica Fı´sica	Dr. Michel Rafat
y Analı´tica	School of Chemistry
Universidad de Oviedo	University of Manchester
33006 Oviedo	Faraday Building, North Campus
Spain	Manchester M60 1QD
[e-mail]	UK
	[e-mail]
Dr. Virginie Pichon-Pesme
Laboratoire de Cristallographie et	Dr. Jesus Rodrı´guez
Mode´lisation des Mate´riaux	Centro de Quimica, IVIC
Mine´raux et Biologiques	Apartado 21827
LCM3B (UMR UHP-CNRS 7036)	Caracas 1020 A
Faculte´ des Sciences et	Venezuela
Techniques	[e-mail]
Universite´ Henri Poincare´
BP 239, Boulevard des	Prof. Bernard Silvi
Aiguillettes	Laboratoire de Chimie The´orique
54506 Vandoeuvre-le`s-Nancy	UMR (CNRS) 7616
CEDEX	Universite´ Pierre et Marie Curie
France	3, Rue Galile´e
[e-mail]-	94200 Ivry sur Seine
nancy.fr	France
	[e-mail]

XXXVIII List of Contributors

Prof. Angelo Sironi

Dipartimento di Chimica Strutturale e Stereochimica Inorganica

Universita` di Milano Via Venezian 21 20133 Milano

Italy

and

CNR – Istituto di Scienze e Tecnologie Molecolari

Via Golgi 19 Milano Italy

[e-mail]

Prof. Miquel Sola`

Institut de Quı´mica Computacional Departament de Quı´mica Universitat de Girona Campus de Montilivi 17071 Girona

Catalonia Spain

[e-mail]

Dr. Nagamani Sukumar

Department of Chemistry and Biological Chemistry

Center for Biotechnology and Interdisciplinary Studies Rensselaer Polytechnic Institute 110 8th St.

Troy, NY 12180-3590 USA [e-mail]

Prof. Vladimir G. Tsirelson

Quantum Chemistry Department Mendeleev University of Chemical Technology

Miusskaya Sq. 9

Moscow 125047 Russia [e-mail] [e-mail]

Dr. David Vega

Departamento de Quı´mica

Facultad Experimental de Ciencias y Tecnologı´a (FACYT)

Universidad de Carabobo Valencia, Edo. Carabobo Venezuela [e-mail]

Prof. Daniel P. Vercauteren

Laboratoire de Physico-Clinic Informatique (PCI)

Faculte´s Universitaires Notre-Dame de la Paix (FUNDP)

Rue de Bruxelles 61 5000 Namur Belgium

[e-mail]

Dr. Tammy L. Welshman

Department of Chemistry University of Manitoba Winnipeg, MB, R3T 2N2 Canada [e-mail]

1

1

An Introduction to the Quantum Theory of Atoms in Molecules

Che´rif F. Matta and Russell J. Boyd

1.1 Introduction

The observation that some properties attributed to atoms and functional groups are transferable from one molecule to another has played a key role in the development of chemistry. This observation provides a basis for group additivity schemes and is exemplified by the constancy of group contributions to thermodynamic and spectroscopic properties. But what is the electronic basis of this empirical transferability? The quantum theory of atoms in molecules (QTAIM) [1], developed by Professor Richard F. W. Bader and his coworkers, relies on quantum observables such as the electron density rðrÞ and energy densities to answer such a question. Other important (related) questions addressed by QTAIM include:

What is an atom in a molecule or a crystal?

How can an atom or a group of atoms be transferable sometimes in very di erent external potentials?

Can one define bonding in molecules unambiguously especially in borderline cases?

This chapter contains a summary of some of the main concepts of QTAIM. A more comprehensive and mathematically elegant treatment can be found in Bader’s book [1].

(Often in this chapter, the word ‘‘molecule’’ includes extended systems such as polymers, weakly bonded molecular complexes, and molecular and ionic crystals, in addition to its more traditional meaning of a single, finite, isolated chemically bonded group of atoms. It will be clear from the context when this term is used in its traditional or in its larger sense.)

1.2

The Topology of the Electron Density

The topology of the electron density is dominated by the attractive forces of the nuclei imparting it with its principal topological feature – a substantial local max-