Atomic units; Molecular Hamiltonian; Born-Oppenheimer ...

1. Atomic Units
Atomic units; Molecular Hamiltonian;
Born-Oppenheimer approximation
The atomic units have been chosen such that the fundamental electron properties are all
equal to one atomic unit. (me=1, e=1, = h/2π = 1, ao=1, and the potential energy in
the hydrogen atom (e 2 /4πε0ao = 1).
Symbol Quantity Value in a.u. Value in SI units
me Mass (Electron mass) 1 9.1094·10 -31 Kg
e Charge (Electron charge) 1 1.6022·10 -19 C
a0
Length (Bohr first radius
( 2
/me 2 ))
In atomic units the mass of a proton (1.6726 x 10-27 kg) would be 1836.15 au, and the
reduced mass of the hydrogen molecule (.503913 g/mol) would be 925.260 au.
The use of atomic units also simplifies Schrödinger's equation. For example the
Hamiltonian for an electron in the Hydrogen atom would be:
SI units:
Atomic units:
Other frequently used energy units:
2
2 1
2 4
m e
0
2
e
r
1 2 1
2 r
1a.u. = 27.212eV = 627.51Kcal/mol = 219470 cm -1 1Kcal/mol = 4.184KJ/mol
Debye: 1ea0 = 2.54181De; 1De=0.3934ea0
1 0.52918·10 -10 m
EH Energy (EH= 2 /mea0 2 ) 1 27.2114eV=4.3597·10 -18 J
Angular momentum 1 1.0546·10 -34 Js
vB Velocity (α·c=e 2 /4πε0 ) 1 2.187691·10 6 m·s -1
τ0 Time (τ0= /EH) 1 2.418884·10 -17 s
4πε0 Vacuum permittivity 1 1.113·10 -10 C 2 /(Jm)
d0 Electric dipole moment (ea0) 1 8.478·10 -30 Cm
e 2 a0 2 EH -1 Electric polarizability 1 1.649·10 -41 C 2 m 2 J -1
Derived units
h Planck’s constant 2π 6.626·10 -34 Js
c Speed of light (c=vB/α) 137.036 2.998·10 8 m/s
μB Bohr magneton (e /2me) ½ 9.274·10 -24 J/T
μN Nuclear magneton 2.723·10 -4
5.051·10 -27 J/T
μ0 Vacuum permeability (4π/c 2 ) 6.629·10 -4 4π·10 -7 Js 2 /(mC 2 )
E0 Electric field strength 1 5.1423·10 11 V/m
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Energy conversion factors
Hartree(a.u.) KJ/mol Kcal/mol eV cm -1
Hartree(a.u.) 1 2625.5 627.51 27.212 219470
KJ/mol 0.00038088 1 0.23901 0.010364 83.593
Kcal/mol 0.0015936 4.184 1 0.043363 349.75
eV 0.036749 96.485 23.061 1 8065.5
cm-1 4.5563E-06 0.011963 0.0028591 0.00012398 1
Other fundamental constants:
Boltzmann’s constant: k=1.38066·10 -23 J/K
Avogadro’s number: NA=6.02205·10 23 mol -1
Rydberg constant: R∞=1.097373·10 7 m -1
Compton wavelength of electron: λC=2.426309·10 -12 m
Stefan-Boltzmann constant: σ=5.67032·10 8 W/(m 2 K 4 )
Electric field unit:
Field=X+a means that an electric field of a*10 -4 a.u. is applied along the X direction.
1.a.u.=Hartree/(charge*bohr) = 27.2114*1.6 10 -19 J/(1.6 10 -19 C * 5.29177 10 -11 m)
1 a.u. = 5.1423 10 11 V/m
1 a.u. = 51.423 V/Å
Ef(V/Å)=51.423Ef(a.u.)
Field=X+1000 means a field of 0.1a.u. (5.1423V/Å) is applied along the X direction
2. Basic concepts
- structure of many-electron operators (e.g. Hamiltonian)
- form of many-electron wave-functions
(Slater determinants, and linear combination of them)
- Hartree-Fock (HF) approximation
- more sophisticated approaches which use the HF method as a starting point
(correlated post-Hartree Fock methods)
Aproximations made in the framework of the Hartree-Fock-Roothaan-Hall theory
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Molecular Hamiltonian
The non-relativistic time-independent Schrödinger equation:
H|Ψ>=E|Ψ>
H – Hamiltonian operator for a system of nuclei and electrons
N
riA
| riA
| |
ri
RA
|
rij
| rij
| |
ri
rj
|
RAB
|
RAB
| |
RA
RB
|
1 2 1 2 ZA
1 ZAZB
i
A
M
A A
r
2 2
i i
A iA r
i j i ij R
1
1
1 1
1 A
1 B
A
AB
T
e
M
T
N
N
M
V
eN
N1
V
N
ee
M1
H
2
i
A molecular coordinate system
2
xi
2
2
y
2
i
2
z
2
i
- the major challenge in solving the time-independent Schrodinger
equation
MA - the ratio of the mass of nucleus A to the mass of an electron
ZA – the atomic number of nucleus A
Te – the operator for the kinetic energy of the electrons
TN – the operator for the kinetic energy of the nuclei
VeN – the operator for the Coulomb attraction between electrons and nuclei
Vee – the operator for the repulsion between electrons
VNN – the operator for the repulsion between nuclei
(1)– represents the general problem
to be separated in two parts: electronic and nuclear problems
Exact solution for systems containing more than one electron is unknown!
approximations, approximations
i, j – electrons (N)
A, B – nuclei (M)
M
V
NN
(1)
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Born-Oppenheimer Approximation
Ψ=Ψ(x1,…,xN, X1,…,XM)
The term VeN in the Hamiltonian prevents any wave-function Ψ(x,X) solution of the time
independent Schrödinger equation from being written as a product of an electronic
wavefunction and a nuclear wavefunction.
Thus, we need approximations!
The nuclei are much heavier than electrons (mproton=1836me)
they move much more slowly
the nuclei can be considered frozen in a single arrangement
(molecular conformation)
the electrons can respond almost instantaneously to any change
in the nuclear position
The electrons in a molecule are moving in the field of fixed nuclei. Neglecting the spin of
nuclei, the total wavefunction is written as:
On this basis:
► 2-nd term in (1) can be neglected
► 5-th term in (1) is a constant
Electronic Hamiltonian:
Ψ=Ψe(x,{R})ΨN(R)
– separable form
- describes the motion of N electrons in the field of M fixed point charges
N
i1
N
M
i1
A 1
iA
N1
N
1 2 ZA
1
He(
R )
i
2
r
(2)
r
i1
ji
He(R) means that He depends on the nuclei positions
Electronic Schrödinger equation:
ij
He(R)Ψe(x;R)=Ee(R)Ψe(x;R) (3)
Ψe=Ψe(x;R) (4)
Ee = Ee(R) (5)
(4) - is the electronic wave-function which describes the motion of the electrons
- describes electronic states for fixed nuclear coordinates {R}
- explicitly depends on the electronic coordinates
- parametrically depends on the nuclear coordinates because H is a function of
the positions R of the nuclei
Another approximation: for each value of the nuclear positions, the electronic system is
in the electronic ground state, corresponding to the lowest Ee(R).
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parametric dependence
- the nuclear coordinates do not appear explicitly in Ψe.
- from the point of view of the electrons, the nuclear degrees of freedom are fixed
- a different electronic wave-function is obtained for each nuclear configuration
The total energy:
M 1 M
pot
ZAZB
Etot ( R ) Ee
(6)
R
A 1 B A
Equations (2) – (6) ≡ electronic problem
The electronic energy is not a constant but depends on the nuclear geometry (molecular
conformation). This geometry dependent electronic energy plays the role of the
potential energy in the Schrodinger equation for the nuclear motion and it is generally
termed potential energy surface (PES).
If the electronic problem is solved
► we can solve for the motion of the nuclei using the electronic energy E(R) as
the potential energy in Schrödinger equation for the nuclear motion.
Since the electrons move much faster than the nuclei
► we can replace the electronic coordinates by their average values (averaged
over the electronic wave-function)
nuclear Hamiltonian
o describes the motion of the nuclei in the average field of the electrons
H
N
M
N
N M N1
N
1 2 1 2 ZA
A i
M
2
r
A A
2
1 i1
i1
A 1
iA i1
j
i
M
A 1
M
A 1
1
2M
1
2M
A
A
2
A
2
A
E
({ R})
E
e
pot
tot
Potential energy surface (PES)
({ R})
M1
M
A 1 B A
Z
A
R
Z
AB
B
pot
Etot
({R})
AB
1
r
ij
M1
M
A 1 B A
Schematic illustration of a potential energy surface
The equilibrium conformation of the molecule corresponds to the minimum of the
potential energy curve
Z
A
R
Z
AB
B
5

nuclear Schrödinger equation
ΨN - nuclear wavefunction
HN|ΨN> = E|ΨN>
o solution of the ro-vibrational problem for the nuclear coordinates, in the
presence of an electronic potential energy surface
o describes the vibration, rotation and translation of a molecule
E - total energy of the molecule (in the Born-Oppenheimer approximation)
- includes: - electronic energy
- vibrational energy
- rotational energy
- translational energy
Total wave-function in the Born-Oppenheimer approximation:
Born-Oppenheimer approximation
- usually a good approximation
- bad approximation for:
Ψ(x,R) = Ψe(x,{R})·ΨN(R)
o excited states (high energy for the nuclear motion)
o degenerate or cuasidegenerate states
Requirements for the wave function
We will assume the Born-Oppenheimer approximation and will only be concerned with
the electronic Schrödinger equation.
1. Normalization
For all quantum mechanical wave-function describing stationary states, is normalized
to unity.
( r) (
r)
dr
1
Integration is performed over the coordinates of all N electrons.
The wavefunction must also be single-valued, continuous and finite.
2. Antisymmetry with respect to the permutation of two electrons
*
Electrons are fermions and a many electron wave-function must be antisymmetric with
respect to the interchange of the coordinate x (both space and spin) of any two
electrons.
Ψ(x1, x2, ... , xi, ..., xj, ...,xN) = -Ψ(x1, x2, ... , xj, ..., xi, ...,xN)
The Pauli exclusion principle is a direct result of this antisymmetry principle.
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3. The electronic wavefunctions must be eigenfunctions of Sz and S 2 operators,
with the eigenvalues MS and S(S+1).
Since the electronic Hamiltonian does not contain any spin operators, it does commute
with the operators Sz and S 2 defined below, with the corresponding eigenvalues MS and
S(S+1), respectively.
S
N
2
sz
MS
S s
z i
i
N
i
2
i
[H,Sz]=0 [H,S 2 ]=0
S(
S
1)
We take care of spin by using spin-orbitals instead of pure spatial orbitals.
α(σ) and β(σ) – spin functions (complete and orthonormal)
(
) (
) d
(
) ( ) d
and
( ) ( ) d
1
1
( ) (
) d
0
0
The electron is described by spatial (r) and spin (σ) coordinates:
Homework
x={r,σ}
1. Write the Z-matrix for the tetramethylsilane
(TMS=Si(CH3)4) molecule in Td symmetry.
2. Use the Gaussian program and obtain the PES of the water molecule in the C2v
symmetry by varying the bond length from 0.9 to 1.1 and the bond angle from 100 o to
110 o . Plot the PES using a dedicated software (Mathematica, MathLab, Mathcad).
3. Write a C program to calculate the
coordinates of the atoms in a graphene sheet
(10x10 atoms) whose structure is given
below. The C-C bond length in graphene is
1.42Å.
The Nobel prize in physics for 2010 was awarded to Andre
Geim and Konstantin Novoselov at the University of Manchester "for groundbreaking experiments regarding the
two-dimensional material graphene".
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