close

Вход

Log in using OpenID

embedDownload
LCAO: The basis set consists of atomic orbitals
Molecular orbitals
(MO‘s)
Atomic orbitals
(AO‘s), 1s, 2s, 2p
Basis sets (VII)
In the literature, there is a number of established basis sets. In the
following some typical examples are shown.
Basis sets of Pople et al. (Gaussian program):
STO-3G
minimum basis (3 GTOs per STO)
3-21G
small “split-valence” basis (double-zeta valence AO)
6-31G*
large “split-valence” basis, polarized at X H
6-311+G**
polarized triple-zeta basis with diffuse functions
Basis sets of Dunning et al.:
DZP
polarized double-zeta basis
TZP
polarized triple-zeta basis
TZ2P triple-zeta basis with 2 sets of polarization functions
TZ2Pf TZ2P + additional f polarization functions
Basis sets (VIII)
Correlation – consistent basis sets:
cc-pVTZ
multiply polarized triple-zeta basis
cc-pVQZ
multiply polarized quadruple-zeta basis
aug-cc-pVQZ cc-pVQZ + additional diffuse functions
RHF/UHF
We have described a HF calculation based on a
Slater determinant like this
spin orbital
Each space orbital is used twice
This is a restricted HF (RHF) calculation
space orbital
RHF/UHF
Another possibility:
spin orbital
space orbital
( )
( )
( )
( )
( )
( )
( )
( )
( )
( )
( )
( )
A new space orbital for each spin orbital
This is an unrestricted HF (UHF) calculation
RHF/UHF
In UHF we have twice as many (for closed shells)
expansion coefficients to vary than in RHF
So, in the HF calculation we willl normally reach a lower
energy in UHF than in RHF
That is good
So why do we not always use UHF?
RHF/UHF
Because UHF has problems with the spin multiplicity:
A = X, Y, Z component of total electron spin:
Square of total electron spin:
The exact electronic wavefunction satisfies:
Possible values of S: 0, ½, 1, 3/2, 2, 5/2, 3, ......
Possible values of MS: -S, -S+1, -S+2, ...., S-1, S
2S+1 possibilities
Multiplicity: 2S+1 – singlet, doublet, triplet, quadruplet .....
RHF/UHF
RHF wavefunction (Slater determinant) has a defined multiplicity and
so it behaves „correctly“:
The UHF wavefunction has no defined multiplicity
If it is just important to get lowest (most accurate)
energy: UHF
If the wavefunction is also important (properties = dipole
moment, ...): RHF
Units
Parenthetic remark: Units
• SI- and cgs-units
• Atomic units
Units
SI and cgs units
Quantity
Mass
Length
Time
Force
Energy
Charge
Units
Principal difference: Units for charge
Q1
Q2
r12
Coulomb energy in SI units:
Coulomb energy in cgs units:
cgs: Two charges, each of 1 e.s.u., separated by 1 cm,
produce a Coulomb energy of 1 erg
Units
Units for dipole moment:
Dipole moment = charge
length
SI: C m
cgs: e.s.u. cm
1 Debye = 1 D = 10-18 e.s.u. cm = 3.33564×10−30 C m
Units
Atomic units, used by MOLPRO
Quantity
units
Mass
Electron mass
Angular momentum
Planck‘s constant/2
Charge
Elementary charge
Length
Energy
Time
Frequency
Momentum
Force
Electrical current
units
Units
Atomic units for dipole moment:
Dipole moment = charge
length
SI: C m
cgs: e.s.u. cm
Atomic units: 1 e a0 = 8.47835281 x 10-30 C m
= 2.54175 D
Koopmans’ theorem
Applications of HF theory
• Koopmans‘ theorem
• Mulliken population analysis
Koopmans’ theorem
Molecule M
Ion M+ (and electron infinitely far away)
How do we get the ionisation energy?
Do HF calculation for M, result EHF(n)
Do HF calculation for M+, result EHF(n-1)
Ionisation energy I = EHF(n-1)
EHF(n)
Koopmans’ theorem
How do we get the ionisation energy in a simpler way?
Do HF calculation for M, result EHF(n)
Calculate HF energy for M+ with same orbitals as
for M, result EHF(M)(n-1)
Ionisation energy I0 = EHF(M) (n-1)
EHF(n)
Some derivation gives Koopmans‘ theorem
Ionisation energy I0 =
j is
j
the orbital energy of the orbital from which the
„ionised“ electron has disappeared
Koopmans’ theorem
Ionisation energies of small molecules (in eV)
Koopmans
HF values from MOLPRO,
Basis set 6-31G**
Geometry optimized at HF
level
Experimental values from
„CRC Handbook of
Chemistry and Physics“,
53rd edition.
Deviations of 10-20%
16.3
Exp.
Koopmans’ theorem
Why is Koopmans relatively successful?
Ionisation energy I0 = EHF(M) (n-1)
EHF(n)
First improvement: Separate HF calculation for M+ lowers
EHF(M) (n-1) and gives lower I0
Second improvement: Introduce Configuration Interaction
(better theory, see later). Lowers both energies, but energy of
M most because M has more electrons. Gives higher I0
So: The two improvements tend to cancel each other and
Koopmans‘ theorem is fairly all right.
Mulliken population analysis
Population analysis???
The molecule has n electrons:
• How many electrons „belong“ to particular nuclei?
• How many electrons are „shared“ between pairs of
nuclei? That is, where are the bonds in the molecule?
Mulliken population analysis
Population analysis
n electrons
[R.S. Mulliken, J. Chem. Phys., 23, 1833, 1841, 2338, 2343 (1955)].
Mulliken population analysis
One electron in one MO.
Differential probability of finding
the electron in the volume element
dV = dx dy dz
Summing over all occupied orbitals gives electron density
Mulliken population analysis
The space orbitals are normalized
and so
n electrons
Mulliken population analysis
In the Roothaan-Hall approximation we have
and so
and
Density matrix
element
Mulliken population analysis
Consequently
and so
That is
n electrons
Mulliken population analysis
Since S
= 1 we can partition the n electrons as follows
n electrons
and
Mulliken interpretation:
P
is the net population of basis function (atomic orbital)
Q is the overlap population of basis functions
and
electrons is shared between these two basis functions
; this fraction of the
Mulliken population analysis
P
is a net population of basis function (atomic orbital)
Q
is the overlap population of basis functions
P33
P22
and
;
Q45
P44
Q12
P11
P66
P55
Mulliken population analysis
Now we can divide the electrons completely between basis functions:
Gross population of basis function (atomic orbital)
Summing over the orbitals gives
n electrons
Mulliken population analysis
We can divide the electrons between nuclei/atoms:
Gross atomic population of the atom A
on
Gross overlap population of the atoms A and B
on
on
Mulliken population analysis
qAB large, positive means „bond“
qAB small, negative means „no bonding“
q2
n electrons
q12
q23
q1
q13
q3
[R.S. Mulliken, J. Chem. Phys., 23, 1833, 1841, 2338, 2343 (1955)].
Mulliken population analysis
H2O
Mulliken values from MOLPRO,
Basis set 6-31G**
Geometry optimized at HF
level
q2 = 8.68
MOLPRO does not seem to be able
to calculate qAB
q1 = 0.66
q3 = 0.66
Mulliken population analysis
O
CH2O+
Mulliken values from GAUSSIAN,
B3LYP/STO-3G
Geometry optimized at B3LYP/STO-3G
level
q12 = 0.45
C
q23 = -0.03
q24 = -0.03
q2 = 7.76
q1 = 4.80
q12 = 0.36
H
q14 = 0.36
q34 = -0.04
q3 = 0.64
H
q4 = 0.64
Mulliken population analysis
CH2O+
GAUSSIAN calculates
=
=
It follows that
+
=

and we get 7.76+4.80+2 0.64+2 0.36+0.45=15.01
1/--pages
Пожаловаться на содержимое документа