Einführung in die Physikalische Chemie: Inhaltepc/huber/PCIpdfs/3-Structure of Molecules.pdf · Einführung in die Physikalische Chemie: Inhalt ... predict the arrangement of atoms

Einführung in die Physikalische Chemie:Inhalt

Einführung in die Physikalische Chemie: Inhalt

Chapter 3: Structure of Molecules

Contents: A- CLASSICAL CONCEPTS:3.1 VSEPR theory3.2 The intramolecular potential3.3 Molecular mechanics (MM) calculations

B- QUANTUM-MECHANICAL CONCEPTS:3.4 Wave-particle duality of light and matter3.5 Introduction to quantum mechanics3.6 Quantum mechanics of atoms3.7 Quantum-mechanical description of molecules

Literature: P. Atkins, J. de Paula, “Atkins’ Physical Chemistry”,8th Ed., Chapters 8, 9, 10, 11I. Tinico et al., “Physical Chemistry, Principles and applicationsin biological sciences”, 4th Ed., Prentice-Hall 2002Chapter 9

Chapter 3: Structure of Molecules

Supplementary Material: Web tutorials “Potential”, “VSEPR-Regeln” and “Molecular Mechanics”

A- CLASSICAL CONCEPTS: 3.1 VSEPR theory

Problem: predict the geometry of molecules

3.1 VSEPR theory

VSEPR (valence-shell electron-pair repulsion) theory is a heuristic concept to predict the arrangement of atoms in a molecule

Procedure:1. Classify the valence electrons of an atom into a) bonding electrons to other atoms b) lone pairs (LPs)2. Determine the number of centres Z: Z=number of neighbouring atoms + LPs3. Distribute the centres Z around the central atom using the following rules: Rule 1: The centres are separated as much as possible Rule 2: LPs are generally larger than the nuclei: if Z=5: LP preferably situated in the equatorial position if Z=6: 2 LPs preferably situated in the axial position

Examples:

Z=2: linear (CO2, BeH2)

Z=3: trigonal planar (BF3, H2CO)

Z=4: tetrahedral (CH4, H2O)

Z=5: trigonal bupyramidal (PF5)

Z=6: octahedral (SF6, BrF6)

Worked example: SO2

Advantagese and disadvantages:+ simple+ allows determination of approximate bond angles (estimate bond lengths from covalent radii)- model is very simplistic and often fails- molecules are not entirely rigid (vibrations, intramolecular motions -> proteins, ...)

3.1 VSEPR theory

3.2 The intramolecular potential

In chapter 2 we discussed intermolecular model potentials to desribe the interaction between molecules in a sample.

In this section we discuss the intramolecular potential and introduce some common model potentials which reflect the interactions between the nuclei and electrons inside a molecule.

As a preparation we must first discuss the concept of motional degrees of freedom of a molecule:

See also the web tutorial “Potential”


3.2.1 Internal and external degrees of freedom of a molecule

Aim: determine the number of independent coordinates (=degrees of freedom, dof) to describe the motions of a molecule.

However, the motions of the atoms inside a molecule are not independent from one another:

Analyse the possible types of motion:

1. An atom can move in all three dimensions in 3D space -> 3 dof

2. A molecule consisting of N atoms can be regarded as a cluster of the

constituent atoms -> 3N dof

• The molecule can move as a hole: Translation -> 3 dof

• The molecule can rotate as a whole: Rotation: linear molecule -> 2 dof

non-linear molecule -> 3 dof

• The remaining 3N-3-2=3N-5 (linear mol.) or 3N-6 (non-linear mol.) dof

account for the internal vibrations of the molecule.


Translation and rotation are external dof because they relate to the motion of the molecule as a whole.

The vibrations correspond to the internal degrees of freedom (coordinates) which are directly related to the molecular structure:

Linear molecules: 3N-5 internal dofNon-linear molecules: 3N-6 internal dof

Hence, the potential energy of a molecule is a function of 3N-5(6) internal coordinates. We speak of a 3N-5(6) dimensional potential-energy surface (PES).

Examples:1. CO2. CO2

3. C2H4

The internal coordinates are frequently expressed in terms of normal coordinates which represent the vibrational modes of the molecule


Example:normal modes of ethene C2H4

3N-6=3x6-6=12 vibrational coords.


3.2.2 The Born-Oppenheimer approximation: potential energy surfaces

The potential energy surface of a molecule depends on both, the coordinates of the nuclei AND the electrons.

However, even the lightest nucleus (H) is 1800 times heavier than an electron. Consequently, the electrons move much faster than the nuclei so that their motions can be adiabatically separated: Born-Oppenheimer-approximation.

The electrons thus always “see” a rigid configuration of the nuclei. When the nuclei move, the electrons instantly rearrange around them to minimise their energy. The potential energy of the molecule thus represents the electronic energy at a given molecular geometry which depends on the 3N-5(6) internal coordinates of the nuclei R1, R2, ...: Epot=V(R1,R2,...): Born-Oppenheimer potential energy surface

A rigorous mathematical formulation is given in PC III (5. Semester, Prof. J.P. Maier)


Example:

The equilibrium geometry of a molecule corresponds to a minimum on the PES.(as the equilibrium distance in a diatomic molecule corresponds to the minimum of the potential-energy curve)

Note: the vibrational coordinates can be determined mathematically by a normal-coordinate analysis at the relevant minimum of the potential energy surface (-> Vorlesung PC III, VTV Spektroskopie).

M.P. Deskevich et al., J. Chem. Phys 124 (2006), Art. No. 224303

2D cut through the PES of the FHCl system relevant for the chemical reaction F+ HCL -> HF + Cl


As a function with 3N-5(6) dimensions is impossible to visualise for N>2, one generally operates with 1D or 2D cuts through the surface.

Repetition: potential energy surfaces (PES)

In the Born-Oppenheimer approximation, the N nuclei of a molecule are held fixed at a geometry described by the internal coordinates R1, R2, ..., R3N-5(6).

The potential energy V of the molecule is then obtained by the calculating the energy of the electrons (solving the quantum-mechanical Schrödinger equation for the electrons, see later) at the given nuclear geometry R1,..,R3N-5(6):V=V(R1, R2, ..., R3N-5(6)).

Internal coordinates: bond lengths rij, bond angles Θijk, dihedral angles α, ...

r12 r13

Θ213

12 3

H

HH

HH

Hα

... or more generally normal (vibrational) coordinates

The potential energy is thus a 3N-5(6) dimensional function (hypersurface) of the internal coordinates R1, R2, ..., R3N-5(6).

Example: the PES of the ozone cation O3+

r12 r13

Θ2132 3

1

Θeq

Potential along the bending coordinate Θ213 (=coordinate of bending vibration)

3N-6=3 internal coordinates 2D contour plot of the potential along the internuclear-distance coordinates r12 and r13.

r 12 /

Å

r13 / Å

Minimum =equilibrium geometry

r12,eq

r13,eq

Excursion: a molecule without structure: CH5+

CH5+ is an important molecule in in chemical-ionisation mass

spectrometry (proton donor !) and in the chemistry of interstellar space

The global minimum of the PES has been computed to be a structure of Cs symmetry consisting of a CH3

+ sub-structure which is linked to a H2 unit: CH3

+ H2

However, the potential energy barriers along reaction paths which lead to an exchange of all hydrogen atoms are extremely low (smaller than the zero-point energy of the molecule !).

CH5+ is thus an extremely fluxional,

structureless molecule which be viewed as a carbon ion which is orbited by five hydrogen atoms !

A satisfactory theory which describes molecules such as CH5

+ has yet to be developed.

3.2.3 1D intramolecular model potentials

(3.1) harmonic

Morse

• most simple potential function to describe intramolecular vibrations

k ... force constantr ... any vibrational coordinate

• good approximation of the potential close to the equilibrium structure req

• becomes unphysical at large r -> no dissociation, no anharmonicities(i.e., deviations from harmonic behaviour)

• compare with classical mechanics of a spring (Hooke’s law):

(3.2)

Harmonic potential:

• the force constant k corresponds to the curvature (2. derivative) of the potential:

(3.3)


• The potential function can be improved systematically by the introduction of anharmonic terms:

harmonic

Morse

• Simple model potential which takes into account anharmonicities and the finite potential well depth

Morse function:

(3.5)De ... dissociation energy

• good representation of the potential function corresponding to stretching vibrations

• not suitable for bending vibrations, torsions

Anharmonic potential function:

(3.4)

-> series expansion of the potential

cubic anharm. quartic anharm.


Internal rotations (torsional motions):

• The potential function for internal rotations is periodic:

Ethane C2H6 Butane C4H10α ... dihedral (torsional) angle

• Torsional potentials can be expressed as sums of trigonometric functions:

(3.6)

• Internal rotations are particular important for the folding of biomolecules (peptides, proteins, RNA, DNA ) and in defining their conformations.

• Definition: a conformation is a specific spatial arrangement of a macromolecule corresponding to a minimum on the PES. Conformers are stereoisomers which can be interconverted by internal rotations (see the examples above).

conformers


3.3 Molecular mechanics (MM) calculations

Molecular dynamics (MD) simulations (see chapter 2.7) are concerned with the modelling of the motion (dynamics) of molecules using inter- and intramolecular interaction potentials

The set of intramolecular model potentials used in MM calculations defines the force field.

Molecular mechanics (MM) calculations are used to explore the potential energy surface (PES) of large molecules by computing the potential energy from model potentials as a function of the internal coordinates. The total potential energy of a molecule is obtained by summing the contributions from all internal coordinates.

MM/MD calculations are used to study the PES and dynamics of large systems (macro- and biomolecules) for which a full quantum-mechanical treatment is not possible.

3.3 Molecular mechanics calculations

3.3.1 Example: the MM2 force field

(compare anharmonic potential function Eq. (3.4)

(compare anharmonic potential function Eq. (3.4)

Θ

(stretch-bend interaction)

N. L. Allinger, J. Am. Chem. Soc. 99 (1977), 8127

(3.7)

(3.8)

(3.9)


3.3.1 Example: the MM2 force field

(compare with Eq. (3.6))

(compare with Eq. (2.3b))

(see chapter 1)

(3.10)

(3.11)

(3.12)


3.3.2 The total potential energy of cyclopropane c-C3H6 calculatedusing the MM2 force field

For simplicity, we disregard the contributions from the H atoms in this example

stretch bend stretch-bend

1 2

3

Etotal depends on r12, r23, r31, Θ123, Θ231 and Θ312. But as the angles Θijk depend on the internuclear distances rij, the potential energy is effectively only a function of three variables.

(3.13)


r13

Θ312

The complexity of the expressions for Etot quickly increase with the size of the molecule.

Example: Combined MM/MD simulation of the folding of the headpiece of the villin protein (time: ca. 10 μs)

Hence, the calculations are usually performed on a computer where the potential energy surface is usually explored using suitable numerical algorithms. A typical task is to compute the local minima on the PES corresponding to stable conformations of molecules.


P.L. Freddolino et al., Biophys. J. 94 (2008), L75

Application example: study the structure and dynamics of biomolecules (protein folding !)

B- QUANTUM-MECHANICAL CONCEPTS:3.4 Wave-particle duality of light and matter

Wave-particle duality: quantum-mechanical objects exhibit properties of both waves and particles

Quantum mechanics: fundamental theory of the microscosm, the physical basis of all chemistry

-> Huygen’s view became the accepted doctrine

Evidence for wave nature of light: diffraction on a grating

(3.14)

separation of lineson grating

diffractionorder

diffractionangle

3.4 Wave-particle duality

History:

• 17th century: Isaac Newton (1643-1727): light is particles(different colours = different particles)

Christian Huygens (1629-95): light is waves(different colours = different frequencies)

Equate Eqs. (3.15) and (3.14):

(3.17)

Where we used

(3.18)λ ... wavelength

momentum

Eq. (3.17) suggests an interpretation of light as a particle (photon) with a momentum p.

Thus, light exhibits the properties of both waves (diffraction) and particles (momentum):wave-matter duality

• 20th century: Max Planck (1858-1947): the energy E of light is quantised:

(3.15) h = 6.626x10-34 Js ... Planck’s constantν ... frequency


Albert Einstein (1879-1955): equivalence of energy and mass:

(3.16) c = 2.9979x108 m s-1 ... speed of light

Thus, light exhibits the properties of both waves (diffraction) and particles (momentum): wave-matter duality of photons

3.4.1 Evidence for the particulate nature of light: photoelectric effect

- electrons are ejected from metal surfaces upon irradiation with UV light- electrons are only ejected if the frequency of the light exceeds a threshold value- the kinetic energy of the electrons scales linearly with the frequency ν above

threshold(3.19)


En

erg

y

Louis de Broglie (1892-1987) generalised this concept to matter: wave-particle duality of matter

(3.20)

3.4.2 Wave nature of matter: the de Broglie relationship

M. Arndt et al., Nature 401 (1999), 680

Evidence for the wave nature of matter: diffraction of C60 molecules

Experiment:diffraction

pattern with grating

no diffraction pattern without

grating


momentumof the particle

velocityof the particle

Compare with Einstein-Eq. (3.17):

(3.17)

3.5 Introduction to quantum mechanics

In quantum mechanics the state of a physical system (atom, molecule, crystal, the universe, ...) is represented by a complex-valued wavefunction ψ(x,t). ψ contains all the necessary information about the system.ψ itself has no specific physical interpretation. However,

(3.21)

is interpreted as the probability P to find the particle in the infinitesimal region of space dx (statistical interpretation of the wavefunction ψ or Kopenhagen interpretation). The results of quantum-mechanical experiments are therefore always interpreted in a statistical sense !

complex conjugate

3.5.1 The wavefunction ψPro memoria: in classical mechanics we solve Newton’s equation Eq. (2.20) to obtain the trajectory x(t) of a particle, i.e., its position as a function of time. Once we know the trajectory, we can calculate all relevant physical quantities of the system, e.g.:

• linear momentum p:

• total Energy Etot:

3.5 Introduction to QM

(3.22)

Since ψ*ψ is a probability, the integral over all space ψ*ψ must be unity. This leads to the normalisation condition of ψ:

3.5.2 Observables, operators and expectation values

Every physical quantity (or observable) O is represented by an operator Ô:

classical mechanics quantum mechanics

• position x position operator (3.23)

• linear momentum p momentum operator (3.24)ħ = h / 2π


• Total energy energy (Hamiltonian) operator

(3.25)kinetic energy

operator

potential energyoperator

In other words: in every measurement of the obsevable Ô we will obtain the same result, namely the eigenvalue o.

Conversely, if the system is in a state with wavefunction ψ which is not a eigenfunction of Ô, we will measure a distribution of different results with a mean value as defined in Eq. (3.27).


The expectation (=mean) value 〈Ô〉of an observable Ô, i.e., the mean result

of an infinite number of measurements of the physical quantity O, is given by:

(3.27)

(3.26)

The value corresponding to a physical quantity O is obtained by acting with the operator Ô on the wavefunction ψ:

numerical value of the physical quantity O= eigenvalue of Ô = a number !!

Mathematically, Eq. (3.26) has the form of an eigenvalue eqution. Thus, ψ is an eigenfunction of Ô and o is the corresponding eigenvalue.

If ψ is an eigenfunction of Ô, then the expectation value exactly corresponds to the eigenvalue o:

(3.28)

Ôψ=oψ(normalisation Eq. (3.22))

3.5.3 The stationary Schrödinger equation

To obtain the stationary (time-independent) wavefunction ψ(x) we have to solve the Schrödinger equation:

(3.29)

one-dimensional time-independent SE

wavefunction potential Energy ħ = h / 2π

Thus, the Schrödinger equation is the eigenvalue equation of the Hamiltonian operator and the stationary wavefunctions are the eigenfunctions of Ĥ.


Alternative formulation:

(3.30)

where H is the Hamiltonian operator

[see also Eq. (3.25)]

kinetic energyoperator

potential energy operator

Summary: principles of quantum mechanics

The quantum-mechanical state of a system is represented by a wavefunction ψ. ψ contains all the information about the system.

ψ has no physical interpretation, but P= ψ*ψ dx is interpreted as the probability of finding the particle in dx (statistical or “Kopenhagen” interpretation of ψ).

The physical quantities (=observables: position, momentum, energy,..) are represented by operators Ô. The value o of a physical quantity is obtained by acting with the operator on the wavefunction:

The wavefunction ψ is obtained by solving the Schrödinger equation of the system (eigenvalue equation for the energy):

(eigenvalue equation)

where H is the Hamiltonian operator

A simple example: the free particle ⇒ blackboard

In classical mechanics, it is possible to determine the value of every physical observable with an arbitrary precision, i.e., all properties of a system can be measured simultaneously.

In quantum mechanics, there are pairs of complementary observables whose values cannot be simultaneously determined with an arbitrary precision, e.g., position x and momentum p. Their uncertainties Δx and Δp must obey the Heisenberg uncertainty relation:

(3.33)

3.6 The Heisenberg uncertainty principle

Where the uncertainty of an observable ΔO is defined by

(3.34)

3.5.4 The Heisenberg uncertainty principle

Illustration of the uncertainty principle: superposition of wavefunctions with a well-defined momentum to generate a wavefunction with well defined position (formation of wavepackets):

ψ

x

Re(ψ)

x

• Wavefunction with a well-defined momentum: ψ=eikx

• Wavefunction with a well-defined position: ψ=δ(x-x0)

δ(x-x0)=Dirac δ function: δ(x-x0)=∞ at x=x0

δ(x-x0)=0 elsewhere

An important consequence is the occurrence of zero-point energy (ZPE) of particles confined to a finite space: because Δx>0 in this case, Δp must be >0 as well. Hence, the particle always has non-zero momentum and therefore non-zero kinetic energy, even at T=0 K !

An analogous relationship holds for the energy E and time t:

(3.35)

3.6 The Heisenberg uncertainty principle

The calculation can easily be generalised to a 3D box:

(3.31)

(3.32)

quantum numbersl,m,n = 1,2,3,...

3.5.5 Another example: the particle in a box

1D problem -> blackboard

The particle in a box represents a model for several important quantum-mechanical systems:

• Quantum-mechanical translational motion

• Quantum dots

• Energy-level structure of molecules with networks of conjugated double bonds


Experimental realisation:scanning tunneling microscope (STM) image of electrons

confined in a 2D box (the quantum corral)

D. Eigler and co-workers, IBM Almaden Research Center

coppersubstrate

ironatoms

density ofsurface

electrons(=ψ*ψ)


3.5.6 Summary: special properties of quantum-mechanical systems

The state of a QM system is represented by a wavefunction ψ, observables by operators Ô.

Certain observables (e.g., position-momentum, energy-time) are complementary and cannot simultaneously assume a well-defined value (Heisenberg uncertainty principle).

When boundary conditions are present, the possible values for physical quantities (energy, momentum, angular momentum ...) are quantised.

The wavefunction is interpreted statistically (P=ψ* ψdx ... probability density).

QM systems confined to a finite region of space exhibit a minimum amount of kinetic energy (zero-point energy).

3.6 Quantum mechanics of atoms

Hydrogen atom: an electron orbiting a proton(the quantum mechanical Coulomb problem)Solve the stationary Schrödinger equation for the problem:

+

-

What is the Hamiltonian operator Ĥ for the problem ?

• Express the classical total energy in terms of momenta px, py, pz and positions x, y, z:

Coulomb potential

(3.36)

• Replace the positions x,y,z and momenta px, py, pz by their operators:

and analoguousfor y, z^ ^

⇒(3.37)

3.6.1 The hydrogen atom


The Schrödinger equation thus becomes:

(3.38)

Mathematical solution: use the symmetry of the problem:the atom has spherical symmetry ⇒ transform to spherical coordinates r, θ, φ:

The single-electron wavefunctions ψ are the hydrogen-atom orbitals.

The wavefunction ψ in spherical coordinates is separable into two functions which only depend on the radial coordinate r and the angular coordinates θ,φ, respectively:

(3.39)

radialfunctions

angularfunctions

(spherical harmonics)hydrogenic

quantum numbers


(3.39)

Hydrogenic quantum numbers:

• n = 1, 2, 3, ... principal quantum number

• l = 0,1,2,3,.., n-1 = s,p,d,f,.. orbital angular momentum quantum number

• m = -l, -l+1, ... 0, ... +l-1, +l magnetic quantum number

The energy levels form a Rydberg series

Energy: only depends on the principal quantum number n:

(3.40)

where:

• RH = 2.1787.10-18 J ... Rydberg constant for the H atomhcRH = 109768 cm-1

... reduced mass of proton and electron

•

Rydberg formula:

Rydberg series

For details of the mathematical solution, see, e.g., Atkins, 8.ed., chapter 10.


The radial wavefunctions Rnl(r):

• number of nodesof the radial functions:nnodes,R = n-l-1

node

• number of nodes of thetotal wavefunction ψ:nnodes = n-1

• the higher the number of nodes, the higher the energy


• The radial distribution function g(R) for the hydrogen atom:

(3.41)

P(r) = g(r) dr represents the probability of finding the electron on the surface of a sphere with thickness dr at the distance r from the nucleus.

1s

2p 3d

2s3p

3s


The angular wavefunctions Ylm(θ,φ) (spherical harmonics):

l = 0 (s):

d=1

l = 1 (p):

d=3

l = 2 (d):

d=5

• number of nodal planes of the angular functions: nnodes,Y = l

nodal plane

• degeneracy d (number of functions with same energy): d=2l+1

pxpypz (m=0) m=±1

dz2 (m=0) dx

2-y

2

dxy dxz dyz m=±1

m=±2


• Interpretation of the quantum numbers: orbital angular momentum

The quantum number m is related to the projection of the orbital angular momentum on the z axis:

operator fororbital angular

momentumalong z axis

(3.44)

Classical angular momentumof a rotating particle:

L= r x mv

It can be shown that the spherical harmonics Yl,m are the

eigenfunctions of the squared angular momentum operator L2 :

(3.42)^

The quantum number l is thus related to the

magnitude L of the orbital angular momentum of the electron :

(3.43)

z

L

l is thus connected with the magnitude of the orbital angular momentum vector,

m with its orientation.


Energy level diagram of the hydrogenic orbitals:

Energy

l

0(s)

1(p)

2(d)

3(f)

n

1

2

3

4

...

• n = 1, 2, 3, ...

• l = 0, 1, 2, 3, ..., n-1

• m = -l, -l+1, ... 0, ... +l-1, +l

The H-atom orbital energiesonly depend on n !Every sub-shell (corresponding to a well-defined value of l) is (2l+1)-fold

degenerate, i.e., it has 2l+1

functions (orbitals) with the same energy.

Shell

K

L

M

N

m=0

m= -1 0 +1

m= -2 -1 0 +1 +2

m=-3 -2 -1 0 +1 +2 +3

Sub-shell:


3.6.2 Many-electron atoms

The configuration of an atom is defined as the orbital occupancies of its electrons, e.g., for hydrogen H (1s)1.

Assume an atom with N electrons with coordinates i where i=1,...,N is a short-hand notation for (ri, θi, φi). In the orbital approximation, the total N-electron wavefunction ψ(1,2,..,N) of the atom is formulated as a product of one-electron wavefunctions which resemble the hydrogenic orbitals:

(3.45)

Generally, within a given shell (=fixed value of n), the larger l, the larger the distance of the

electron from the nucleus, the lower the effective nuclear charge.


The building-up principle

Because the electrons in lower shells are closer to the nucleus, they shield the nuclear charge acting on the electrons in the outer shells (shielding). The outer electrons thus experience a reduced effective nuclear charge.

1s

2p 3d

Radial distribution functionsfor H orbitals

Electrons with a high value of l see a reduced nuclear charge

⇒ reduced Coulomb attraction

⇒ higher energy

⇒ energy of non-hydrogenic atomic orbitals depends on both n AND l.

Energy

l

0(s)

1(p)

2(d)

3(f)

n

1

2

3

4

...

• n = 1, 2, 3, ...

• l = 0, 1, 2, 3, ...

• m = -l, -l+1, ... 0, ... +l-1, +l

5

Building-up principle

In the ground-stateconfiguration of an atom, the electrons occupy the orbitals

with the lowest energies. Every orbital can hold two paired

electrons.

Reason:see the following example


The building-up principle: examples

He: N=2

• We first need to consider that electrons possess an intrinsic angular momentum (the spin) with quantum number s=1/2. The spin angular momentum vector can assume two orientations:

- Spin up ≡ α spin ≡ ms=1/2 ≡ ↑- Spin down ≡ β spin ≡ ms = -1/2 ≡↓

where the spin magnetic quantum number ms indicates the orientation of the spin vector.

• In the ground state of He, both electrons occupy the 1s orbital: He (1s)2


• Hence, the electrons in an atom can be described with four different quantum numbers: n, l, m, ms

• Fermions are elementary particles with half-integer spin: s=1/2, 3/2, 5/2, ...

• Electrons (s=1/2) are fermions !

• 2-electron wavefunction for He → blackboard

• The single-electron wavefunction of electron number i in the atom can thus be represented as a product of a spatial function ψnlm (as defined by Eq. (3.39))

and a spin function χs: (3.46)

where for α spin

where for β spin

• The total N-electron wavefunction ψ(r1,..,rN) must obey the Pauli principle:

The total wavefunction must change signwhen the coordinates of two fermionic particles

are exchanged:

(3.47)


• From this discussion immediately follows the Pauli exclusion principle:

In an atom no two electrons can have the same values for all quantum numbers.

• For He this means that if both electrons occupy the 1s orbital, i.e., n1=1, l1=0, m1=0

and n2=1, l2=0, m2=0

they must differ in the spin magnetic quantum number ms:ms,1=1/2 and ms,2=-1/2

• This is often represented graphically as follows:

1s

• The configuration of He is thus denoted as: He (1s)2

orbital occupationnumber


Li: N=3

• Configuration: Li (1s)2 (2s)1 1s

2s

C: N=6

• Configuration: C (1s)2 (2s)2 (2p)2

1s

2s

2p Hund’s maximum multiplicity rule:

An atom in its ground state adopts the configuration with the greatest number of unpaired atoms.

(reason: electron repulsion)


3.7 Quantum-mechanical description of molecules:molecular orbital theory

Similar to the situation in atoms, one can approximately factorise the molecular wavefunction ψmol into a product of one-electron molecular wavefunctions (molecular orbitals, MOs).

3.7 Quantum mechanics of molecules

The molecular orbitals can be approximatively constructed by a linear combination of atomic orbitals (LCAO).

Example: the hydrogen molecule H2:

(3.48)

Linear combination of two H 1s atomic orbitals:

σg σu

Nomenclature:• σ ... cylindrical symmetry around

intermolecular axis

• g ... gerade: symmetric under inversionu ... ungerade: antisymmetric under inversion

H(1) H(2)r r

H(1) H(2)

Energy level diagram:

energy is lowered compared to free atoms

bonding MO:

energy is raised compared to free atoms

antibonding MO:

En

erg

y


R1s(r)

Example: the oxygen molecule O2:

• O: 8 electrons per atom: O (1s)2 (2s)2 (2p)4

• LCAO of the 1s and 2s orbitals form pairs of σg/u orbitals like in H2.

• LCAO of the 2p orbitals forms a pair of σg/u and πg/u orbitals:

LCAO of pz orbitals(along intermolecular axis)

LCAO of px,y orbitals(perpendicular to

intermolecular axis)

πg

πu


For revision and better understanding:

Web-Tutorials “Potential” and “Molecular Mechanics”

Sheet “Lernziele 3” to be downloaded from the website

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Einführung in die Physikalische Chemie: Inhaltepc/huber/PCIpdfs/3-Structure of Molecules.pdf · Einführung in die Physikalische Chemie: Inhalt ... predict the arrangement of atoms