LQTA
Introduction to MOLCAS
1. Introduction to HF and electron correlation
The Hartree Fock (HF) approximation would provide the lowest expectation value to
HF energy if the expansion basis set was complete. However, the use of an infinite
number of basis functions is, computationally, unaffordable. The bigger the basis set,
the more flexible becomes the wavefunction and more accurate is the calculation.
Nevertheless, once a certain number of basis functions has been reached, the energy
does not improve if this number is increased. This set can be considered as complete,
and the energy it provides is the HF limit.
The correlation energy is defined as the difference between the exact non-relativistic
energy of the system and the HF limit:
!!!!!!!!!!!!!!"## = ! ℇ! − ! !! !!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!
HF does not include electron correlation since the electron-electron repulsion is treated
in an average way. This treatment considers that each electron moves in an average
potential arising from the N-1 electrons. For a sufficiently large basis set, HF accounts
for 99% of the total energy. The remaining 1% may be important for describing
chemical problems.
1. Introduction to HF and electron correlation
Two types of electron correlation:
• Due to the mono configurational nature of the HF method, systems where more
than one configurations are important cannot be accurately described. In this
sense, HF is not able to account for non dynamical (or static) correlation. The
most popular way to account for this part of the correlation is to explicitly include
more than one Slater determinant to expand the trial wavefunction while at the
same time optimizing the orbitals in the SCF procedure. Static correlation can
become really important in distorted molecules when bonds are created or
broken. It is also essential for treatment of open-shell molecules like excited states
or transition metals complexes.
• Dynamic correlation arises from the rij-1 term in the Hamiltonian operator. One
electron, in the HF method, is supposed to move in an average potential created
by the rest of the electrons. However, the movement of the electrons is correlated,
when one electron moves the rest are affected.
1. Introduction to HF and electron correlation
Example: H2 dissociation energy
The shape of the RHF
and “exact”curves are
very different,
specially at large H-H
distances.
Why?
2. Introduction to Multiconfigurational Based Methods
1σu
At the equilibrium distance, the H2
wavefunction can be described as a H H
single determinant where the two H H
electrons are placed at the 1sg bonding 1sA 1sB
orbital
Ψ = a0φ HF 1σg
However, at large distance, the possibility of finding different electron distributions
is larger:
Ψ = a φa + b φb
H H
H H
b
a € H+ H-
H H
2. Introduction to Multiconfigurational Based Methods
How we describe these systems?
The starting point is usually HF and, then, several determinants can be generated by
moving electrons from occupied to unnocuppied orbitals. This leads to Slater
determinants that are single, double, triple … excitations. If all possible determinants in
a given basis set are included, all the electron correlation is recovered.
… …
HF Single Exc. Double
(S) Exc. (D)
2. Introduction to Multiconfigurational Based Methods
Configuration Interaction (CI)
We can write the wavefunction as a linear combination of Slater determinants:
Ψ = a0φ HF + ∑ aiφ i
i=1
Then, the expansion coefficients a are obatined by minimizing the energy. The molecular
orbitals used for the excited Slater determinants are taken from a HF calculation and
fixed.
€
Ψ = a0φ HF + ∑ aS φ S +∑ aD φ D + ∑ aT φT +...
S D T
A full CI wavefunction means that we include all possible determinants. However, this
is computationally unafordable and the number of excited determinants in the CI
expansion must be truncated.
2. Introduction to Multiconfigurational Based Methods
MultiConfigurational Based Methods
The Multi-Configuration Self-Consistent Field (MCSCF) method represents a flexible
solution to handle special chemical problems (bond breaking, quasi-degenerate states,
surface crossings…) where more than one slater determinant is important. It can be
considered as a CI where both the coefficients in front of the determinants and the
MOs used for constructing the determinants are optimized.
The MCSCF wavefunction is a truncated CI expansion, being one of the main problem
of this method to select the configurations that must be included.
Ψ!"#"$ = !! !"# !
!
Configuration State Functions
Coefficient
in each configuration
for each configuration
Both are Optimized
3. Introduction to CASSCF
MultiConfigurational Based Methods
The most widely used MCSCF method is the Complete Active Space Self-
Consistent Field (CASSCF) approach.
This method is based on the division of the orbitals into subsets (primary and
secondary orbitals) depending on their occupation at the MCSCF wavefunction:
1. Primary space 2. Secondary space
• Active orbitals can have • also called external or virtual,
any occupation between 2
and 0 electrons, while • orbitals remain unoccupied (0).
• Inactive orbitals are always
doubly occupied (2).
3. Introduction to CASSCF
MultiConfigurational Based Methods
CASSCF (N,n)
A common notation for this procedure is CASSCF(N,n), which denotes the number
of electrons (N ) and orbitals (n ) contained in the active space (AS).
The CASSCF wavefunction is built with all the possible configurations involving
the active electrons and the active orbitals. In other words, a full CI is performed
with the active orbitals and all the resulting determinants are included in a MCSCF
calculation, recovering static correlation
CASSCF (10,10) : 10 electrons in 10 orbitals
CASSCF (14,12) : 14 electrons in 12 orbitals
3. Introduction to CASSCF
CASSCF (N,n)
CASSCF
Inactive
Primary space
FCI! Active (RAS2)
Secondary space Secondary
3. Introduction to CASSCF
CASSCF (N,n)
CASSCF, importance of the Active Space
Single Point Calculation for the Ground State
2500000 (16,14)
(12,10)
Number of CSFs’
(10,8)
2000000
(8,6)
1500000
1000000 (14,12)
500000
0
0 200 400 600 800 1000 1200
Time (min)
3. Introduction to CASSCF
CASSCF (N,n)
3. Introduction to CASSCF
How we select the active space?
There are several general rules that help to select an appropriate active space for a given
molecule:
RULE 1. If an occupied orbital correlates with a virtual orbital, both should be
included in the active space.
If we πco π*co
include
this
bonding
orbital … We should
also include
the
corresponding
antibonding
orbital …
3. Introduction to CASSCF
How we select the active space?
There are several general rules that help to select an appropriate active space for a given
molecule:
RULE 2. Orbital energies can be another criteria to select orbitals.
Occupied orbitals with the highest energy and virtual orbitals with the lowest
energy must take part in the active space.
RULE 3. The choice of the orbitals can also be made according to their
occupations.
Orbitals with zero or two occupation numbers should not be considered as part of
the active space. The most important orbitals to include are those with occupation
numbers between 1.98 and 0.02.
3. Introduction to CASSCF
How we select the active space?
There are several general rules that help to select an appropriate active space for a given
molecule:
RULE 4. Orbitals that are essential to describe the chemical problem cannot be
excluded from the active space. IMPORTANT in our project!
Example, if we
want to study the
O-O bond
Dissociation
All Orbitals
involving the
O-O bond have
to be included in
the AS
3. Introduction to CASSCF
How we select the active space?
Practical HINTS:
1. Based on the above rules select an
initial active space.
2. Perform several trials
increasing/decreasing the active space
to test the dependence of the result on
the active space chosen.
Some example
3. Introduction to CASSCF
How we select the active space?
Example: Acrolein
Complete ! O
space would contain:
hν
4 electrons distributed in:
-------------------------------- H
2 ! bonding orbitals (2e- each)
2 !* anti-bonding orbitals (0e- each)
H H*
There is
also one lone
pair of O (2e-).
This give an active space
of 6 electrons in 5 orbitals
3. Introduction to CASSCF
How we select the active space?
15 is in this case the
Example: Acrolein Highest Occupied
Nº orbital (from SCF calculation!) O O Molecular Orbital =HOMO
hν O
H +
.
H
16 is the Lowest
Inactive . H*
Unnocupied
H
Molecular
H H*
12 Orbital=LUMO
310-254 nm 250 nm
13 !CO (2e)
14 LP O (2e) So our active space of 6
15 !CC (2e) electrons in 5 orbitals
Active involves from the orbital
------
16 !CO*(0e) number 13 to the orbital
17 !CC* (0e) number 17.
18
See below in the STEP 2
Secondary 19 input (CASSCF
.
. calculation) .. The last
. orbital in our AS is 13 so
the number of inactive
orbitals is 12.
3. Introduction to CASSCF
How we select the active space?
Example: Acrolein
O O
hν O
H +
H
H H* H* H
310-254 nm 250 nm
Hypothesis 1. We want to describe the dissociation of the C=O double bond. Is the
6,5 active space suitable?
No! We should try to include all the orbitals involved in the C=O bond, the
pCO/pCO* orbitals are included in the 6, 5 active space but the sCO/sCO* are not.
We should increase the active space to 8,7 in order to well describe the double bond.
sCO sCO*
This increase of the active space
change the number of inactive
orbitals …
3. Introduction to CASSCF
CASSCF (N,n)
Static correlation
CASSCF are rarely used for calculating large fractions of correlation energy. Orbital
relaxation does not usually recover much electron correlation. It is much more
efficient to include additional determinants and to keep the MOs fixed (CI). MCSCF
usually generate good wavefunctions that recover the static part of the correlation.
Dynamic correlation
CASSCF does not include dynamic correlation. However, it can be included through
Multi-Reference Configuration Interaction (MRCI) or using Second order
perturbation theory (CASPT2). This is equivalent to use CI or Pertubation theory
(MP2) to incorporate electron correlation to HF wavefunctions but in this case,
instead, we use MCSCF wavefunctions as a reference.
4. Introduction to MOLCAS
http://www.molcas.org
Back to MOLCAS …
is an ab initio quantum chemistry software package that allows:
(1) Accurate description of general molecular problems in the ground and excited
states.
(2) The treatment of more specific problems where a single determinant is not enough
and requires a multiconfigurational approach.
(3) Optimizations, Minimum Energy Paths, Intrinsic reaction coordinates, Transition
states ….
4. Introduction to MOLCAS
Flowchart
…has a modular design…
Each calculation is achieved
by executing the specific module
4. Introduction to MOLCAS: SEWARD and WATEWAY INPUT
STEP 1 [SCF.in] The first thing that MOLCAS does is to read the information contained
in the GATEWAY module (basis set and geometry). Then we compute the integrals
using the &SEWARD and the HF wavefunction with &SCF.
allows to use molden to visualize orbitals
* user comments
& to specify the module you want to use
molecular geometry is taken from file: min.xyz
specify basis set and
symmetry group if any
computes the integrals
computes the Hartree-Fock wave
function
4. Introduction to MOLCAS: CASSCF INPUT
STEP 2a. [CASSCF.in] We look to the SCF orbitals obtained in STEP 1, we decide an
adequate active space to prepare the input below and we run then the CASSCF
calculation:
INPUT: Add after the &GATEWAY and SEWARD modules the following:
The calculation needs a set of orbitals as guess could be from a previous SCF
calculation (or CASSCF).
last inactive orbital out of the active
space
n, number of orbitals in the active space
number of states to compute (1: in this case
we are only interested in the ground state)
and highest state to be computed (1)
4. Introduction to MOLCAS: CASSCF INPUT and OUTPUT
STEP 2b. [CASSCF.out]
When the CASSCF is finished we have to look to the orbitals with molden to check if the
active space is correct. If not, we can use the ALTER keyword within the RASSCF
module to exchange pairs of orbitals.
In the example we are changing 1 pair of orbitals of symmetry 1
And these orbitals are 11 and 15…
We can take the CASSCF energy and see which determinants are the most important
ones from our OUPUT:
!CO (2e) !CO (2e)
LP O (2e) LP O (2e)
!CC (2e) !CC (0e)
----------- Double
Exc. (D) -----------
!CO*(0e) !CO*(2e)
!CO*(0e) !CC->!CO* !CO*(0e)
4. Introduction to MOLCAS: Optimization INPUT
STEP 3. Optimize the ground state.
INPUT:
compute the gradients (analytic or numerical) requested by
alternative modules
generates the new geometry
defines the largest change allowed for the internal coordinates
maximum number of iterations which will be allowed.
the loop ends when the geometry converges or when
maximum number of iterations (MaxIter) is reached
module necessary to calculate vibrational frequencies
TS (Transition State search)
&SLAPAF IRC (Intrinsic Reaction Coordinate)
possibilities MEP-search (Minimum Energy Path)
Constrained optimizations
4. Introduction to MOLCAS: Optimization INPUT
STEP 4. We can correct the CASSCF energies introducing dynamic correlation by
performing CASPT2 calculations.
INPUT: Add after the &RASSCF section:
Shift applied to the denominator
to avoid intruder states. It can be real or imaginary.
RLXRoot Specifies which root to be relaxed in a geometry optimization of a
Frozen multi state CASPT2 wave function
This keyword is used to specify the number of frozen orbitals, i.e. the
orbitals that are not correlated in the calculation.
Specifies the states to include into the
multi state calculation:
5 which are 1 2 3 4 5
5. LQTA Exercise
Compute the dissociation curve for a given CFC’s molecule at the SCF,
CASSCF and CASPT2 level:
(1) Take the optimized minimum optimized with Gaussian. Perform an SCF
calculation. Have a look to the orbitals.
(2) Choose an appropriate active space and perform the CASSCF calculation.
Check with MOLDEN that the active space is the desired one.
(3) Run constrained optimizations along the most probable dissociation channel
at the SCF and CASSCF levels. Plot the dissociation curves at both levels of
calculation.
(4) Obtain CASPT2 energies by performing single point calculations at selected
points of the dissociation curve.
