Ideas of Quantum Chemistry P72 potx

The Molecule in an Electric or Magnetic FieldMAGNETIC PHENOMENA • An elementary particle has a magnetic dipole moment M proportional to its spin angular momentum I, i.e.. 12.82, in which

676 12 The Molecule in an Electric or Magnetic Field

MAGNETIC PHENOMENA

• An elementary particle has a magnetic dipole moment M proportional to its spin angular momentum I, i.e M= γI, where γ stands for what is called the gyromagnetic factor (characteristic for the kind of particle)

• The magnetic dipole of a particle with spin I (corresponding to spin quantum number I) in homogeneous magnetic field H has 2I+ 1 energy states EmI = −γmI¯hH, where

mI= −I −I + 1    +I Thus, the energy is proportional to H

• The Hamiltonian of a system in an electromagnetic field has the form

ˆ

j =1

 1 2mj



ˆpj−qi

cAj

2 + qjφj

 + ˆV

where Ajand φj denote the vector and scalar fields at particle j (both are functions of position in the 3D space) that characterize the external electromagnetic field

• A and φ potentials contain, in principle (see Appendix G), the same information as the magnetic and electric field H andE There is an arbitrariness in the choice of A and φ.

• In order to calculate the energy states of a system of nuclei (detectable in NMR spec-troscopy) we have to use the Hamiltonian ˆH given above, supplemented by the inter-action of all magnetic moments related to the orbital and spin of the electrons and the nuclei

• The refinement is based on classical electrodynamics and the usual quantum mechanical rules for forming operators (Chapter 1) or, alternatively, on the relativistic Breit Hamil-tonian (p 131) This is how we get the HamilHamil-tonian (12.66) which contains the usual non-relativistic Hamiltonian (12.67) plus the perturbation (12.68) with a number of terms (p 657)

• Experimentalists use an empirical Hamiltonian (eq (12.82)), in which they have the interaction of the nuclear spin magnetic moments with the magnetic field (the Zee-man effect), the latter weakened by the shielding of the nuclei by the electrons plus the dot products of the nuclear magnetic moments weighted by the coupling con-stants The experiment gives both the shielding (σA) and the coupling (JAB) con-stants

• Nuclear spin coupling takes place through the induction mechanism in the chemical bond (cf Figs 12.13, 12.15) Of key importance for this induction is high electron density at the position of the nuclei (the so called Fermi contact term, Fig 12.13)

• The theory of shielding and coupling constants was given by Ramsey According to the theory, each quantity consists of diamagnetic and paramagnetic contributions The diamagnetic term is easy to calculate, the paramagnetic one is more demand-ing

• Each of the contributions to the shielding constant individually depends on the choice

of the origin of the vector potential A, while their sum is invariant with respect to this choice

• The London atomic orbitals χL= exp(−iAC · r)χ(r − RC) used in calculations for

a molecule in a magnetic field depend explicitly on that field, through the value AC

of the vector potential A calculated at the centre RC of the usual atomic orbital χ(r− RC)

• The most important feature of London orbitals is that all the integrals appearing in calcu-lations are invariant with respect to the origin of the vector potential This is why results obtained using London orbitals are also independent of that choice

Main concepts, new terms

Hellmann–Feynman theorem (p 618)

Cartesian multipole moments (p 624)

dipole, quadrupole, octupole moments

(p 624)

multipole polarizability (p 628)

multipole hyperpolarizability (p 628)

induced dipole moment (p 628)

ZDO (p 635)

sum over states method (p 635)

finite field method (p 639)

Sadlej relation (p 640)

second/third harmonic generation (p 646)

nuclear magnetic dipole (p 648)

spin magnetic moment (p 648)

gyromagnetic factor (p 648)

Bohr magneton (p 648)

nuclear magneton (p 648)

Maxwell equations (p 962)

NMR (p 658) NMR Hamiltonian (p 658) shielding constants (p 659) spin–spin intermediate coupling (p 659) local field (p 659)

chemical shift (p 664) Ramsey theory (p 666) diamagnetic effect (p 668) paramagnetic effect (p 668) coupling constant (p 668) direct spin–spin interaction (p 669) diamagnetic spin–orbit contribution (p 669) paramagnetic spin–orbit (p 670)

spin–dipole contribution (p 670) Fermi contact contribution (p 670) coupling mechanism (p 672) London orbitals (p 673) GIAO (p 673)

From the research front

The electric dipole (hyper)polarizabilities are not easy to calculate, because:

• the sum over states method (SOS) converges slowly, i.e a huge number of states have to

be taken into account, including those belonging to a continuum;

• the finite field method requires a large quantity of atomic orbitals with small exponents

(they describe the lion’s share of the electron cloud deformation), although, being diffuse,

they do not contribute much to the minimized energy (and lowering the energy is the only

indicator that tells us whether a particular function is important or not)

More and more often in their experiments chemists investigate large molecules Such

large objects cannot be described by “global” polarizabilities and hyperpolarizabilities

(ex-cept perhaps optical properties, where the wave length is often much larger than size of

molecule) How such large molecules function (interacting with other molecules) depends

first of all on their local properties We have to replace such characteristics by new ones

offering atomic resolution, similar to those proposed in the techniques of Stone or Sokalski

(p 1018), where individual atoms are characterized by their multipole moments,

polariz-abilities, etc

Even a few years ago, the shielding and especially spin–spin coupling constants were

very hard to calculate with reasonable accuracy Nowadays these quantities are computed

routinely using commercial software with atomic London orbitals (or other than GIAO basis

sets)

The current possibilities of the theory in predicting the nuclear shielding constants and

the nuclear spin–spin coupling constants are shown in Tables 12.1 and 12.2 Note that the

accuracy of the theoretical results for shielding constants is nearly the same as that of

ex-periment As to the spin–spin coupling constants, the theoretical results are only slightly off

experimental values

678 12 The Molecule in an Electric or Magnetic Field

Table 12.1. Comparison of theoretical and experimental shielding constants The shielding constant

σA(unitless quantity) is (as usual) expressed in ppm, i.e the number given has to be multiplied by

10 −6to obtain σ

A of eq (12.83) The Hartree–Fock, MP2, MP4 results are calculated in J Gauss,

Chem Phys Letters 229 (1994) 198; the CCSD(T) in J Gauss, J.F Stanton, J Chem Phys 104 (1996)

2574, and the CASSCF in K Ruud, T Helgaker, R Kobayashi, P Jørgensen, K.L Bak, H.J Jensen,

J Chem Phys 100 (1994) 8178 For the Hartree–Fock method see Chapter 8, for the other methods

mentioned here, see Chapter 10 The references to the corresponding experimental papers are given in

T Helgaker, M Jaszu´nski, K Ruud, Chem Rev 99 (1999) 293 The experimental error is estimated for

σHin ammonia as ±10, for σ O as ±172, for σ H in water as ±0015, for σ F as ±6, for σ H in hydrogen fluoride as ±02

Table 12.2. Comparison of theoretical and experimental spin–spin coupling constantsnJABfor eth-ylene (n denotes the number of separating bonds), in Hz For the methods used see Chapter 10 All references to experimental and theoretical results are in T Helgaker, M Jaszu´nski, K Ruud, Chem.

Rev 99 (1999) 293

Spin–spin coupling constants JABfor ethylene, in Hz Method 1JCC 1JCH 2JCH 2JHH 3JHH-cis 3JHH-trans

Ad futurum .

It seems that the SOS method will be gradually sent out of business The finite field method (in the electric field responses) will become more and more important, due to its simplicity

It remains however to solve the problem, how to process the information we get from such computations and translate it into the above mentioned local characteristics of the molecule Contemporary numerical methods allow routine calculation of polarizability It is diffi-cult with the hyperpolarizabilities that are much more sensitive to the quality of the atomic basis set used The hyperpolarizabilities relate to non-linear properties, which are in high demand in new materials for technological applications

Such problems as the dependence of the molecular spectra and of the molecular con-formations and structure on the external electric field (created by our equipment or by a neighbouring molecule) will become more and more important

The theory of the molecular response to an electric field and the theory of the molecular response to a magnetic field, despite some similarities, look as if they were “from another story” One of the reasons is that the electric field response can be described by solving the Schrödinger equation, while that corresponding to the magnetic field is based inherently

on relativistic effects The latter ones are much less investigated except some quite simple

examples Another reason may be the scale difference: the electric effects are much larger

than the magnetic ones

However, the theory for the interaction of matter with the electromagnetic field has to

be coherent The finite field method, so gloriously successful in electric field effects, is in

the “stone age” stage for magnetic field effects The propagator methods84 look the most

promising, these allow for easier calculation of NMR parameters than the sum-over-states

methods

Additional literature

A.D Buckingham, Advan Chem Phys 12 (1967) 107.

A classical paper on molecules in a static or periodic electric field

H.F Hameka, “Advanced Quantum Chemistry Theory of Interactions between

Mole-cules and Electromagnetic Fields”, Addison-Wesley Publishing Co., Reading,

Massa-chusetts, USA (1965)

This is a first class book, although it presents the state of the art before the ab initio

methods for calculating the magnetic properties of molecules

T Helgaker, M Jaszu´nski, K Ruud, Chem Rev 99 (1999) 293.

A competent review article on the magnetic properties of molecules (NMR) with

pre-sentation of suitable contemporary theoretical methods

Questions

1 The Hellmann–Feynman theorem says that ( ˆH means the Hamiltonian depending on

the parameter P):

a)∂E∂P= ψ|∂ ˆ H

∂P|ψ, if ψ is the variational trial function;

b) ∂E∂P = ψ|∂ ˆ H

∂P|ψ, if ψ is the Hartree–Fock wave function;

c) ∂E∂P= ψ|∂ ˆ H

∂P|ψ, for any normalized ψ;

d) ∂E∂P = ψ|∂ ˆ H

∂P|ψ, if ψ is an eigenfunction of ˆH

2 The proportionality constant at the third-power term (the powers of electric field

in-tensity) in the expression for the energy of a molecule in a homogeneous electric field

is:

a) a component of the quadrupole moment;

b) a component of the dipole moment;

c)−1

3 !β, where β denotes a hyperpolarizability component;

d) a component of the octupole moment

3 A non-polar molecule (with a non-zero quadrupolar moment) in an electric field with a

non-zero gradient:

a) does not interact with the field;

b) will rotate to align its dipole moment along the field;

c) will orient to align its longer quadrupole axis along the field;

d) will orient to align its longer quadrupole axis along the gradient of the field

84J Linderberg, Y Öhrn, “Propagators in Quantum Chemistry”, 2nd edition, John Wiley & Sons, Ltd,

2004.

680 12 The Molecule in an Electric or Magnetic Field

4 Second harmonic generation requires that the molecule has:

a) large dipole hyperpolarizabilities;

b) large quadrupole and octupole polarizabilities;

c) large octupole and higher order hyperpolarizabilities;

d) a large quadrupole hyperpolarizability

5 In variational calculations for the benzene molecule (zero electric field) the GTO expo-nents and positions have been optimized A small shift of the GTOs when using a finite field method:

a) will always lower the energy;

b) always increases the energy;

c) will increase the energy if the GTOs move in the direction of the field and decrease

if they move in the opposite direction;

d) will always give a polarizability greater than zero

6 The magnetic moment M of a particle:

a) always has the direction of the particle’s spin angular momentum;

b) its length is always an integer or half-integer;

c) interacts with a homogeneous magnetic field H , and the interaction energy is equal

to12MH2; d) interacts with a homogeneous magnetic field H , and the interaction energy is equal

to−H · M

7 If we choose the vector potential A(r)=1

2[H × r], where H is the magnetic field inten-sity, then:

a) we have∇A = 0 and A agrees with the Maxwell equations;

b) A(r) is a homogeneous field;

c) A(r) is directed towards the origin;

d) A(r) is parallel to H

8 The vector potential A(r) of electromagnetic field corresponds to homogeneous mag-netic field H Then A:

a) is uniquely determined from the Maxwell equation;

b) is uniquely determined from the Maxwell equation A= curl H;

c) is also a homogeneous field;

d) curl[A − ∇(x2+ y3+ z4)] = H

9 A nuclear shielding constant consists of diamagnetic and paramagnetic parts Each of these parts:

a) represents a second-order effect in perturbation theory;

b) represents a first-order effect in perturbation theory;

c) changes when the origin of the vector potential A changes;

d) represents the Fermi contact term

10 The London or Gauge-Invariant Atomic Orbital χL(r− R):

a) depends on the vector potential calculated at position R;

b) depends on the vector potential calculated at position r;

c) does not depend on the vector potential;

d) depends on the vector potential at the point shown by r− R

Answers

1d, 2c, 3d, 4a, 5c, 6d, 7a, 8d, 9c, 10a

Chapter 13

I NTERMOLECULAR

I NTERACTIONS

Where are we?

We are already in the crown of the TREE

An example

Why does liquid water exist? Why do molecules stick together at low temperatures? Visibly

they attract each other for some reason The interaction is not however very strong since

water evaporates when heated (without destroying the water molecules)

What is it all about

THEORY OF INTERMOLECULAR INTERACTIONS

Interaction energy concept ( ) p 684

• Natural division and its gradation

• What is most natural?

Dissociation barrier ( ) p 687 Supermolecular approach ( ) p 689

• Accuracy should be the same

• Basis set superposition error (BSSE)

and the remedy

• Good and bad news about the supermolecular method

Perturbational approach ( ) p 692

• Intermolecular distance – what does it mean?

• Polarization approximation (two molecules) ()

• Intermolecular interactions: physical interpretation

• Electrostatic energy in the multipole representation and the penetration energy

• Induction energy in the multipole representation

• Dispersion energy in the multipole representation

Symmetry-Adapted Perturbation Theories (SAPT) ( ) p 710

• Polarization approximation is illegal

• Constructing a symmetry adapted function

• The perturbation is always large in polarization approximation

681

682 13 Intermolecular Interactions

• Iterative scheme of the symmetry adapted perturbation theory

• Symmetry forcing

• A link to the variational method – the Heitler–London interaction energy

• When we do not have at our disposal the ideal ψA 0and ψB 0

Convergence problems ( ) p 721 Non-additivity of intermolecular interactions ( ) p 726

• Many-body expansion of interaction energy

• Additivity of the electrostatic interaction

• Exchange non-additivity

• Induction energy non-additivity

• Additivity of the second-order dispersion energy

• Non-additivity of the third-order dispersion interaction

ENGINEERING OF INTERMOLECULAR INTERACTIONS

Van der Waals surface and radii ( ) p 742

• Pauli hardness and the van der Waals surface

• Quantum chemistry of confined space – the nanovessels

Synthons and supramolecular chemistry ( ) p 744

• Bound or not bound

• Distinguished role of the electrostatic interaction and the valence repulsion

• Hydrogen bond

• Coordination interaction

• Hydrophobic effect

• Molecular recognition – synthons

• “Key-and-lock”, template and “hand-and-glove” synthon interactions

Chapter 8 dealt with the question of why atoms form molecules Electrons and nuclei attract each other, and this results in almost exact neutralization of matter Despite this, atoms and molecules interact, because

• two atoms or molecules cannot occupy the same space,

• electrons and nuclei in an atom or molecule may still interact with those in other atoms

or molecules

This chapter will tell us about the very reason for this and will give details of the interac-tion

Why is this important?

What is the most important message humanity ever learned about matter? According to

Richard Feynman the message would be: “The world is built of atoms, which repel each other

at short distances and attract at longer ones” If the intermolecular interactions were suddenly

switched off, the world would disintegrate in about a femtosecond, that is in a single period

of atomic vibration (the atoms simply would not come back when shifted from their equilib-rium positions) Soon after, everything would evaporate and a sphere of gas, the remainder

of the Earth, would be held by gravitational forces Isn’t it enough?

What is needed?

• Perturbation theory (Chapter 5, absolutely)

• Variational method (Chapter 5, recommended).

• Appendix X, p 1038 (absolutely)

• Many-Body Perturbation Theory (MBPT) (Chapter 10, p 554, necessary)

• Reduced resolvent (Chapter 10, p 554, necessary)

• Appendix Y, p 1050 (recommended)

• Appendix T (mentioned)

Classical works

Such an important subject was recognized

very early The idea that the cohesion of

matter stems from the interaction of small

indivisible particles (“atoms”) comes from

Democritus. An idea similar to that cited

by Feynman was first stated clearly by the

Croat scientist Rudjer Boskovi´c in “Theoria

Philosophiae naturalis”, Venice, 1763. Padé

approximants were first proposed in the PhD

thesis of Henri Padé entitled “Sur la

représen-tation approchée d’une fonction pour des

frac-tions rationnelles”, which was published in

Annales des Sciences d’Ecole Normale

Su-perieure, Suppl [3], 9 (1892) 1. The role

Democritus of Abdera (ca.

460 B.C – ca 370 B.C.), Greek philosopher, founder

of the first atomic theory Ac-cording to him, nature repre-sents a constant motion of in-divisible and permanent par-ticles (atoms), whose interac-tions result in various materi-als.It turned out after almost

25 centuries that this hypoth-esis was basically correct!All the written works of Democri-tus have been lost, but his

ideas continued to have an important impact on science for centuries.

of intermolecular interactions was

highlight-ed in the work of Johannes Diderik van

der Waals, especially in “Die Kontinuität des

gasformigen und flüssigen Zustandes”, Barth,

Leipzig (1899, 1900) From that time on,

intermolecular interactions are often called

van der Waals interactions. The concept

of ionic radii was first proposed by Linus

Pauling in “The Sizes of Ions and the

Struc-ture of Ionic Crystals”, Journal of the

Ameri-can Chemical Society, 49 (1927) 765. The

Rudjer Josip Boskovi´c (1711–

1787), a Croat physicist, math-ematician, astronomer and philosopher from beautiful Dub-rovnik.

quantum mechanical explanation of

inter-molecular forces, including the ubiquitous

dispersion interactions, was given by Fritz

London in “Zur Theorie und Systematik der

Molekularkräfte”, Zeitschrift für Physik, 63

(1930) 245 and in “Über einige Eigenschaften

und Anwendungen der Molekularkräfte” from

Zeitschrift für Physikalische Chemie (B), 11

(1930) 222. Linus Pauling, invited Baker

Lecturer to Cornell University, wrote one of

the most seminal books in chemistry “The

Nature of the Chemical Bond”, Cornell Univ.

Press, Ithaca, 1948, where inter alia he

eluci-dated the role of hydrogen bonds in forming

Johannes Diderik van der Waals (1837–1923), Dutch physicist, professor at the University of Amsterdam His research topic was the influ-ence of intermolecular forces

on the properties of gases (equation of state of the real gas, 1873) and liquids In

1910 van der Waals received

a Nobel Prize “for his work

on the equation of state for gases and liquids”.

structures. The hydrophobic effect was first

highlighted by Walter Kauzmann in a paper “Some Factors in the Interpretation of Protein

684 13 Intermolecular Interactions

Denaturation”, in Advances in Protein Chemistry, 14 (1959) 1, the effect was further

elabo-rated by George Nemethy, Harold Scheraga, Frank Stillinger and David Chandler among others. Resonance interactions were first described by Robert S Mulliken in an article

“The Interaction of Differently Excited Like Atoms at Large Distances”, in Physical Reviews,

120 (1960) 1674. Bogumił Jeziorski and Włodzimierz Kołos extended the existing

the-ory of intermolecular forces to intermediate distances (“On the Symmetry Forcing in the

Perturbation Theory of Weak Intermolecular Interactions”, International Journal of Quantum Chemistry, 12 Suppl 1 (1977) 91).

THEORY OF INTERMOLECULAR INTERACTIONS

There are two principal methods of calculating the intermolecular interac-tions: the supermolecular method and the perturbational method Both assume the Born–Oppenheimer approximation

13.1 INTERACTION ENERGY CONCEPT

The idea of interaction energy is based on the Born–Oppenheimer (clamped nu-clei, see eq (6.4)) approximation Let us define interaction energy at the configu-ration R of the nuclei as

Eint(R)= EABC(R)−EA(R)+ EB(R)+ EC(R)+ · · · (13.1) where EABC(R) is the electronic energy (corresponding to E0(0)from eq (6.21))

of the total system, and EA(R), EB(R), EC(R), are the electronic energies of

the interacting subsystems, calculated at the same positions of the nuclei as those in

the total system

13.1.1 NATURAL DIVISION AND ITS GRADATION

Although the notion of interaction energy is of great practical value, its theoretical meaning is a little bit fuzzy Right at the beginning we have a question:

interac-tion of what? We view the system as composed of particular subsystems, that once

isolated, then have to be put together

For instance, the supersystem

may be considered as two interacting water molecules, but even then we still have

an uncertainty, whether the two molecules correspond to (I) or to (II):

In addition the system might be considered as composed of a hydrogen molecule

interacting with two OH radicals:

etc

Choice of subsystems is of no importance from the point of view of mathematics,

but is of crucial importance from the point of view of calculations in theoretical

chemistry

The particular choice of subsystem should depend on the kind of experiment

with which we wish to compare our calculations:

• we are interested in the interaction of water molecules when studying water

evaporation or freezing;

• we are interested in the interaction of atoms and ions that exist in the system

when heating water to 1000◦C

Let us stress, that in any case when choosing subsystems we are forced to

sin-gle out particular atoms belonging to subsystem1 A and B It is not sufficient to

define the kind of molecules participating in the interaction, see our examples I

and II

If when dividing a system into n subsystems in two ways (I and II), we obtain

|Eint|I<|Eint|II, division I will be called more natural than division II. natural

subsystems

13.1.2 WHAT IS MOST NATURAL?

Which division is most natural? We do not have any experience in answering such

questions What? Why should we have any difficulties? It is sufficient to consider

all possible divisions and to choose the one which requires lowest energy

Unfortu-nately, this is not so obvious Let us consider two widely separated water molecules

(Fig 13.1.a)

1 This means that the interaction energy idea belongs to classical concepts In a quantum system,

particles of the same kind are indistinguishable A quantum system does not allow us to separate a part

from the system Despite this, the interaction energy idea is important and useful.

interacting with two OH radicals:

etc

Choice of subsystems is of no importance from the point of view of mathematics,

but is of crucial... the

Perturbation Theory of Weak Intermolecular Interactions”, International Journal of Quantum Chemistry, 12 Suppl (1977) 91).

THEORY OF INTERMOLECULAR INTERACTIONS...

• Quantum chemistry of confined space – the nanovessels

Synthons and supramolecular chemistry ( ) p 744

• Bound or not bound

• Distinguished role of the electrostatic