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```QuantumChemistryandSpectroscopy
Text book Engel: Quantum Chemistry and Spectroscopy (3 ed., QCS) or
Engel and Reid: Physical Chemistry (3 ed.)
Chapter8.Quantumchemistry(Engel:QCSchapter15)
Let us finally concentrate to actual quantum chemistry. The
Hamiltonian is known
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Z I e2
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2
m
4
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|
r
R
|
4
pe
r
iI
j <i
I >J 4pe0 | RI - RJ
êë i
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In this we need to know the atomic type and positions. The
positions do not need to be exact but they need to be
reasonable. The molecular geometry can be optimized later. As
said earlier the exact wave functions cannot be solved and thus
some approximations are needed. We also utilize the variational
principle to find the best trial wave functions from the chosen
function class. We already used non-self consistent LCAO trial
functions but they are too simple.
Hartree-Fock equations
The simplest anti-symmetric product function is the Slater
determinant
Y ( r1 , r2 ,.., rN ) =
1
det | j 1 ( r1 )j 2 ( r2 )..j N ( rN ) |
N!
Where the φ(r) is an atomic type orbital. We need to write there
using some basis functions ξ(r)(simple known functions that
contain adjustable parameters. Typically the basis functions are
centered to atoms, ξ(r-R).
( ) =
( )
,
Now we can insert to the Slater determinant and the basis
functions to variation equation (or to Schrödinger equation).
After a bit of math we can write the Roothaan-Hall (R-H)
equation
=
Where C contain all the molecular coefficients, F is the Fock
matrix and S is the overlap matrix. The Fock matrix is
(
=∫
| )=
( ) −
=
( )
ħ
(
=
∇ −
+
| ) ∑
( ) ( ) ( )
| − |
=
( )
|
−
=
|
( )
(
( )
|
)
=
,
,
This is quite complex equation since the solution of the R-H
equation is hidden to the coefficients J and K. Also the
integrals (nm|kl) contain 4 functions (and they are 6
dimensional). The J can be simplified but K not. If there is M
basis functions the computations scale as M4. These together are
the Hartree-Fock (HF) equations. They cannot be solved directly.
One need to make a guess of C(0) and solve K(0), J(0) and F(0)
matrixes with this guess. Then the R-H equation can be solved
and a new set of coefficients C(1) can be solved. Usually one
have to adjust the new C’s a bit but this self-consistent loop
usually converges quite well (at least if there is a large HOMOLUMO gap).
Solve K(n) J(n) and H
Guess
C(0)
Solve F(n)C’(n+1)=S C’(n+1)
Is DC(n)small
enough
Get new C(n+1)= (1-a)C(n)
+
aC’(n+1)
The next issue is the basis functions, ξ(r). There are several
possibilities but the most common functions are gaussians
( )=
( )exp(−
)
( , )
The Pn(r) is some polynomial to make the functions more atomic
like and Ylm are the spherical harmonic functions. It would be
nice to use atomic type functions (Slater functions) but the J
and K integrals become tedious with them. With gaussians these
integrals can be done analytically but we need more Gaussians to
get good accuracy. It is close to an art to make a good Gaussian
basis. Typically we need higher angular momentums than the
valence electrons have, e.q. d-orbital for C, O, N etc. We need
two (or even more) exponents since in the molecule the wave
functions decay from the nucleus is not symmetric.
Polarization functions: basis functions with higher angular
momentum
Double-zeta (DZ), triple-zeta (TZ) functions: basis functions
with different exponents
Diffuse functions: very broad gaussians, needed for intermolecular interactions.
In summary the HF theory and the basis functions will limit
the accuracy of the calculations. If the basis is very good and
in practice do not cause any practical error to the calculations
the results are referred to be at the HF limit. With modern
computers and rather small molecules it not difficult to get to
the HF limit. But even then the results are not very good since
the HF itself is not very accurate. Well some quantities, like
bond distances, are very good with HF but for example the
binding energies are not.
The geometries are good but most bonds are a bit too short.
We need to discuss a bit how to optimize the molecules geometry
using the HF theory. We can compute the total electronic energy
of the system. That depend of the atomic positions. Then we can
compute the forces acting on each atom
(
,..
);
=−
Once we have the forces several minimization algorithms can be
used to minimize the molecules geometry. We can easily find only
the nearest minima from the starting geometry. Often the
molecule will have several local minima. In simple molecules we
can what the minima might be. Unfortunately the number of minima
will increase rapidly when the size of the molecule increases.
There is not general method to estimate the number of minima of
a relatively large molecule but it is very high. For this reason
any computational research of proteins
is difficult or impossible if we do not
have a good guess of the structure. To
some extent this is true for solid
materials too but still for most
molecules we are interested in
chemistry this is not a (big) problem.
Also the vibrational frequencies are not easy. In fact they are
difficult quantities to compute since they are based of very
small energy differences. The vibrations can be computed as the
second derivative of the total energy
=
From this matrix we need to find the vibrational normal
coordinates and the vibrational eigenvalues. The computations
are rather technical but easy to do. The most demanding part is
the calculation of the second derivative.
In fact any property related to molecules structure can be
computed. One very interesting field
is chemical reactivity. Reactions
equilibrium constant and reaction
rate can be computed (but they are
not very accurate). The reactants
will have a certain lowest energy
geometry, the transition state will
have an other geometry and the
products will again have some
geometry. If these geometries can be
found the Reactant, Transition state
and Product energies can be
computed. Unfortunately the relevan
energy is not the electron energy
but the free energy, F = H – TS, so
we need to estimate the entropy. In
gas phase molecules and on surfaces
this is easy (we need the molecular
vibrations) but it is not easy in
liquid. In this course we will not go to the details.
Beside the structural porperties also the electronic properties
can be computed. Of these one very interesting one is the dipole
moment. The change of the dipole moment will determine the
strength of the vibration adsorption peaks. The HF dipole
moments are not very accurate but they are systematically
overestimated.
Any other electronic quantity can be computed, like
polarizations. Maybe surprisingly the electronic energy states
and especially the exited states are not very reliable in HF.
Summary: HF is the basis of all traditional quantum chemical
methods. It is not considered to be very accurate and in any
serious quantum chemical calculations some “post HF” should be
used.
Post Hartree-Fock methods
There are many ways to improve the HF method. Most of them are
very technical and in this course only the basic ideas behind
them are given. But we can first “define” the correlation
energy. There is the best non-relativistic total energy with
Born-Oppenheimer approximation of the system Etot, the
correlation energy is the difference
Ecorr = Etot - EHF < 0
This is not very useful definition but if we have different well
defined methods to compute the correlation energy, the one which
have lowest energy is the best.
The conceptually simplest post HF method is the Configuration
Interaction (CI) methods. In this method the wave functions is
built from several deteminants
Ψ( , . .
)=
Ψ +
,
Ψ +
,
Ψ +. ..
The new determinants are built form the HF orbitals but the
electrons are also placed on exited states. The notation Ψ
means that one electron is excited from state i to state a,
similarly Ψ
means that two electrons are excited from states i
and j to states a and b. This wave functions is denoted as CISD
(CI with singlet and double excitations). The Ψ is the Slater
determinant (or the HF wave functions). All the excited
determinants are orthogonal to the HF wf (and each other). In
the CI calculations the HF orbitals are kept constant and the a
coefficients are optimized.
The CISD method is a reasonable solution for the correlation
energy for small molecules but for larger molecules it become
inefficient. Also for larger molecules CISD is not very accurate
and higher terms would be useful but methods like CISDT become
computationally very expensive. There will be a huge amount of
excitations to be compute. In general the pure CI type methods
become too expensive when the molecules size in increasing
Scaling
Method
N4
HF
N5
MP2
N6
MP3, CISD, CCSD
N7
MP4, CCSD(T)
N8
(MP5), CISDT, CCSDT
N10
(MP7), CISDTQ, CCSDTQ
The CI has also a size consistency problem. This can be
illustrated with an example of He dimer. The He atom have only
two electrons and thus CISD is an exact method for it. When the
He2 is studied at CISD level the theory is not exact since the
triple and quadrupole excitation are missing.
In general finite-CI accuracy reduce with the size of the
molecule and any calculation which deals with molecules
association is biased with the size consistency. This is a
rather large huge problem for computational binding energies and
an unfortunately feature for many Post-HF methods.
There are several variations to the CI method, like Coupled
Cluster (CC) and multireference (MRCI) methods but they are too
complex for this course.
Perturbation methods
Another approach to correlation it the perturbation methods. The
idea is very general and it can be used to several other quantum
chemical problems. We can assume that we can solve Hamiltonian
H(0) and the full Hamiltonian have a small perturbation V. The
perturbation will be scale with parameter l.
( )
=
+
We can next write the energy and wave functions as a power
series of l.
=
( )
+
( )
( )
+
+⋯
( )
=
+
( )
( )
+
+⋯
Now we can write hierarchical equation for each power of l
( )
( )
( )
+
( )
+
( )
( )
( )
( )
=
=
( )
=
( )
( )
( )
+
( )
( )
+
( )
( )
( )
+
( )
( )
( )
The different level wave functions are orthogonal
So we can solve
( )
=〈
( )
( )〉
( )
, Ψ( ),
( )
=〈
( )
( )
( )
etc.
( )〉
( )
=∑
〈
( )
( )
( )〉
( )
( )
( )
=
.
( )
=
〈
( )
( )
−
( )〉
( )
The perturbation theory can be applied to correlation energy.
This is called Moller-Plesset theory (MPn). Then the H (0) is the
HF theory and the perturbation is
1
=
−
−
(
−
1
2
)
This clever since it corrects the HF error. The level (2)
correction is
( )
=
,
[( | ) − ( | )]
+ − −
The i,j denote occupied state and a,b unoccupied states. This
the first real correction and it is called the MP2 theory. (The
E(0) + E(1) is in fact the HF energy). The perturbation theory can
be to arbitrary level. The different levels are denotes as MPn.
Of these the most important is the MP2. It is rather fast and
causes a significant correction to HF. The MP3 is a
disappointment since it does not corrects much the MP2. The MP4
is somewhat better but it is already very expensive. The higher
terms are not used and in some model systems it has been shown
that the MPn series do not converge.
Computationally the MP2 with small approximations can be
implemented very effectively. The RI-MP2 method is much faster
than and essentially as accurate as true MP2. All practical MP2
calculations for larger systems are done with RI-MP2 (or its
variants). In big RI-MP2 calculations the time consuming part is
the HF.
The perturbation method is very general and it can used in
several quantum chemical problems. One application is the CI
(and CC) methods where the higher order excitations can be
solved with perturbation theory. The CCSD(T) method is the
“golden standard” of quantum chemistry and it has the same
scaling behavior as MP4.
Last the basis set needed for well converged CI and MPn methods
is larger than in the HF calculations. This is bad news since
these post-HF methods have poorer scaling behavior than HF so
the computations become quickly very time consuming.
Density Functional Theory
The traditional Quantum Chemical methods will approach
systematically the solution of the correlation energy but they
lead to very heavy computational procedures. The Density
Functional Theory will utilize another approach. In its hart is
a proof that the ground state of an electronic system depend
only on the electron density.
Y ( r1 , r2 ,.., rN ) a Y [n ( r ) ]
This wave function is much much simpler than the true wf. The
problem is that we do not know the equations to solve the wf.
The basic ideas are form Kohn, Hohenberg and Sham (1963, 1964,
Kohn Nobel prize 1998). Even we do not know the correct equation
it is useful to try to find a good approximation for it. Kohn
and Sham showed that a single Slater determinant type wf with
simple model for the Exchange and Correlation (XC) energy will
give good results. The formulation is quite similar as for the
HF.
=∫
=
( ) −
=
ħ
=
∇ −
=
+
∑
|
( ( ), ∇ ( ) , . . )
( ) ( )
| − |
−
( )
|
;
( ) = =
,
,
( )
( )
The difficult part is the XC energy. It is not known but several
approximation are done. The simplest Local Density Approximation
(LDA) depend only on density, EXC,LDA[n(r)]. This functional can
be obtained from exact calculations of homogenous electron gas.
The LDA work reasonably well and it is better than HF (with few
exceptions) but electrons are NOT homogenously spread in
molecules. The next level is the Generalized Gradient
Approximation, EXC,GGA[n(r),∇ ( )]. The GGA models are much better
but there is no simple (or even non-simple) procedure to build
the GGA functionals. For this reason there are a huge amount of
different GGA models. The most common are BLYP and PBE. Even
there are far too many GGA models, the DFT-GGA method have one
very strong advantage. It scales as N3. It is much more accurate
than HF but it is much faster. With DFT we cannot get systematic
improvement of the accuracy but it can be applied to big systems
are the results are almost always good (same level as MP2).
One can also mix the HF and DFT theories. These are called
hybrid methods and they also are rather accurate. The most
common of them is the B3LYP model also PBE0 model is good.
As DFT cannot be improved systematically the results matters.
The DFT wf’s are variational but the models are not, so we
cannot judge the quality of results with the correlation energy.
The results can be compared to experiments or to more accurate
quantum chemical methods.
Below there are few tables.
They show that B3LYP method is usually as good as MP2. Even the
tables shows only one DFT method the general trend is correct.
In most cases the DFT methods are as good as or better than MP2.
There are more tables in Engels book. Almost all methods the
bond distances are quite good. For energies the HF or LDA are
not reliable whereas most GGA’s and correlated quantum chemistry
methods are good. The reaction barriers are challenging form
computational methods and in them the GGA methods work rather
well.
Naturally the GGA methods have some problems. One of them is the
van der Waals interactions (or dispersion) which is important in
weakly bonded systems. Basically HF and GGA will do not have
terms. In most cases such PBE-D3 models works well. DFT have
also problems to describe the hydrogen correctly. Like HF the
DFT models do not give good HOMO-LUMO gap.
There is a lot of research focusing on development of the
computational methods. In traditional quantum chemistry one area
is on development fast computational algorithms for the known
methods. In DFT the development is rather slow. Now there is
quite a bit of interest of beyond-DFT methods. Like RPA, MP2
etc.
On the other hand both these methods have enormous amount of
applications covering almost all fields of chemistry. DFT can be
used to model single molecules, clusters of atoms, surfaces,
solid systems and liquids. Systems up to 100.000 valence
electrons can be computed. An emerging research field is quantum
mechanical materials screening where properties of materials (or
molecules) will be evaluated computationally and this
information is used to help the synthetic work.
```
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