Optimizing Geometries
=====================

.. : minima, transition states, crossings, and minimum energy paths

It is now useful to explore potential energy surfaces (PES) and optimize the molecular geometry for
specific points along the PES. Different cases are discussed including a way to obtain the optimal geometry
in a minimum energy search, to obtain a transition-state structure connecting different regions of
the PES, to find the crossing between two PES where the energy becomes degenerate, or to map
the minimum steepest-descent energy path (MEP) from an initial point to the final
a minimum energy geometry as the PES progresses in a downward manner.

All these types of searches can be performed either by fully optimizing all
degrees of freedom of the system or by introducing certain restrictions. |molcas| can perform
geometry optimizations at the SCF (RHF and UHF), DFT (RHF and UHF based), CASSCF (CASSCF and RASSCF) levels of theory,
where efficient analytical gradients are available and at the CASPT2 and other correlated levels where numerical
gradients are used.

Geometry optimizations require many cycles, in which the electronic energy is estimated at a specific
level of calculation followed by calculation of the gradient of the energy with respect to the geometric
degrees of freedom (DOF). With this information at hand, the program must decide if the molecule is
already at the final required geometry (i.e. gradient :math:`\sim` 0 for all
DOF) indicating a minimum in the PES or if the geometry must be modified
and continue the cycle. The input file should,
therefore, be built in a way that allows a loop over the different programs.

The general input commands :command:`Do while` and :command:`Enddo` control the loop
and program input is inserted within these commands. Instructions for the number of maximum iterations allowed and the type of output required can also be added.
(see :numref:`UG:sec:sysvar`)

.. The commands :command:`Set output file`, which prints output for each iterations and
   in the :file:`$WorkDir` directory with the file name :file:`Structure.$iteration.output`, and
   :command:`Set maxiter 100`, which sets maximum iterations to one hundred.

All examples previously discussed, use :kword:`COORD` keyword, but it also possible
to use *native format*, where symmetry unique atoms are specified (:kword:`SYMMETRY`)
and provide generators to construct all atoms in the molecule.

The selected example describes geometry optimization of the water molecule at the SCF RHF level
of calculation:

.. extractfile:: problem_based_tutorials/Water_distorted.xyz

  3
   coordinates for water molecule NOT in equilibrium
  O 0.000000  0.000000  0.000000
  H 0.758602  0.000000  0.504284
  H 0.758602  0.000000 -0.504284

.. extractfile:: problem_based_tutorials/SCF.minimum_optimization.H2O.input

  *SCF minimum energy optimization for H2O
  *File: SCF.minimum_optimization.H2O
  *
  &GATEWAY
   Title= H2O minimum optimization
   coord=Water_distorted.xyz
   basis=ANO-S-MB
   group=C1

  >>> Do while
   &SEWARD ;&SCF; &SLAPAF
  >>> EndDo

The sequence of programs employed includes :program:`GATEWAY` which is external to the loop, followed by
:program:`SEWARD`, :program:`SCF`, and :program:`SLAPAF`. :program:`SEWARD`
computes the integrals, :program:`SCF` program computes the RHF energy and wave
function, and :program:`SLAPAF` will control the calculation of gradients and
estimate if the calculation has already finished or needs to proceed to a new
nuclear geometry for the next iteration. Automatically, a file named
:file:`$Project.geo.molden` will be generated in :file:`$WorkDir` containing all the
geometric steps contained in the optimization process. :program:`MOLDEN` or :program:`LUSCUS` can
then read this file to display the individual molecular geometries which form the optimization cycle.

Using another reference wave function can be simply performed by changing the sequence of
programs. For instance, we can perform an UHF calculation of the :math:`\ce{H2O^+}`
cation:

.. extractfile:: problem_based_tutorials/UHF.minimum_optimization.H2Oplus.input

  *UHF minimum energy optimization for H2O+
  *File: UHF.minimum_optimization.H2Oplus
  *
  &GATEWAY
   Title= H2O minimum optimization
   coord=Water_distorted.xyz
   basis=ANO-S-MB
   group=C1
  >> Do while

   &SEWARD
   &SCF; Title="H2O minimum optimization"; UHF; Charge=1
   &SLAPAF

  >> EndDo

The same procedure can be followed if we pretend to perform a DFT geometry optimization:

.. extractfile:: problem_based_tutorials/DFT.minimum_optimization.H2O.input

  *DFT minimum energy optimization for H2O
  *File: DFT.minimum_optimization.H2O
  *
  &GATEWAY
   Title= H2O minimum optimization
   coord=Water_distorted.xyz
   basis=ANO-S-MB
   group=C1

  >>> Export MOLCAS_MAXITER=100
  >>> Do while

   &SEWARD
   &SCF ; Title="H2O minimum optimization"; KSDFT=B3LYP
   &SLAPAF &END

  >>> EndDo

Once an energy minimum is found based on the calculation of gradients, it is necessary to
ensure that the geometry really is a minimum energy point. This can be only
accomplished by computing second derivatives of the energy (i.e. the Hessian).
|molcas| can compute analytical Hessians for SCF and single state
CASSCF wave functions. For other methods, numerical procedures can be used
to compute the Hessian. Once the Hessian is computed, vibrational
frequencies are calculated, and Statistical Mechanics is used to obtain thermodynamic
properties. At a true energy minimum, there will be :math:`3N-6` real frequencies
Program :program:`MCKINLEY` computes second derivatives
of a predefined (SCF or CASSCF) wave function, while :program:`MCLR` performs
the vibrational and statistical analyses. |molcas| simply requires input for
the :program:`MCKINLEY` program to perform the entire calculation by using keywords
:kword:`Perturbation` and :kword:`Hessian`, while program :program:`MCLR` will be
called automatically but requires no input.
The full set of calculations is included below first a geometry optimization followed by the
calculation of a Hessian.

.. extractfile:: problem_based_tutorials/SCF.minimization_plus_Hessian.H2O.input

  *SCF minimum energy optimization plus hessian of the water molecule
  *File: SCF.minimization_plus_hessian.H2O
  *
  &GATEWAY
   Title= H2O minimum optimization
   coord=Water_distorted.xyz
   basis=ANO-S-MB
   group=C1

  >>> Export MOLCAS_MAXITER=100
  >>> Do while

   &SEWARD
   &SCF; Title="H2O minimum optimization"
   &SLAPAF

  >>> EndDo

  &MCKINLEY

Note that :program:`MCKINLEY` input above is placed after :command:`EndDo`, and, therefore,
is external to the looping scheme. Once the geometry optimization at the desired level of theory has finished, the
Hessian will be computed at the final geometry.
In general, any calculation performed using a :file:`$WorkDir` directory where a
previous geometry optimization has taken place will use the last geomtry calculated
from that optimization as the input geometry even if :program:`SEWARD` input is
present. To avoid that, the only solution is to remove the communication file
:file:`RUNFILE` where the geometry is stored. Note also, that the frequencies are
computed in a cartesian basis, and that three translational and three rotational
frequencies which should be very close to zero are included in the output file.
This is not the case when numerical gradients and Hessians are used.
In particular, for water at its minimum energy structure three (:math:`3N-6`)
real vibrational frequencies. By default, in :file:`$WorkDir` a file :file:`$Project.freq.molden`
is generated containing the vibrational frequencies and modes, which can be visualized by :program:`MOLDEN`.

A new level of theory, CASSCF, is introduced here which is especially suited for
geometry optimizations of excited states discussed in the next chapter.
A geometry optimization is performed to illustrate a broader range of possibilities including
the imposition of a geometric restrain that the HOH angle in water should be constrained to 120\ |o|
during the optimization.
This means that only the O--H bond distances be optimized in this partial minimization.
The restriction is indicated
in in :program:`GATEWAY`
by invoking the keyword :kword:`Constraints` and ending with the keyword :kword:`End of Constraints`.
The names of variables corresponding to geometrical variables in either internal or Cartesian coordinates
that are to be constrained are placed between these two keywords.
(see nomenclature in
:numref:`UG:sec:definition_of_internal_coordinates`)
In the case of :math:`H_2O`, the H1--O--H2 angle is fixed at 120\ |o|, so a variable,
:math:`a`, is first defined with the keywork :kword:`Angle`, which relates it to the H1--O1--H2 angle, followed by the second keyword, :kword:`Value`,
where the variable :math:`a` is specified as 120\ |o|.
It is not required that the initial geometry is 120\ |o|, only that the final result for the calculation
will become 120\ |o|.

Note that the :program:`RASSCF` program requires initial trial orbitals, and those
which are automatically generated by :program:`SEWARD` are used. The resulting CASSCF
wave function includes all valence orbitals and electrons.

.. extractfile:: problem_based_tutorials/CASSCF.minimum_optimization_restricted.H2O.input

  *CASSCF minimum energy optimization of the water molecule with geometrical restrictions
  *File: CASSCF.minimum_optimization_restricted.H2O
  &GATEWAY
   Title= H2O minimum optimization
   coord=Water_distorted.xyz
   basis=ANO-S-MB
   group=C1
  Constraint
     a = Angle H2 O1 H3
    Value
     a = 90. degree
  End of Constraints

  >>> Do while

   &SEWARD
   &RASSCF; nActEl=8 0 0; Inactive=1; Ras2=6
   &SLAPAF

  >>> EndDo

Other more flexible ways to impose geometric restrictions involve the specification of which internal
coordinates should remain fixed and which should change. In the next example,
the bond lengths are forced to remain fixed at their initial distance (here 0.91 Å), while the
bond angle, having an initial of 81\ |o|, is optimized.

.. extractfile:: problem_based_tutorials/DFT.minimum_optimization_restricted.H2O.input

  *DFT minimum energy optimization of the angle in the water molecule at fixed bond lengths
  *File: DFT.minimum_optimization_restricted.H2O
  *
  &GATEWAY
   Title= H2O minimum optimization
   coord=Water_distorted.xyz
   basis=ANO-S-MB
   group=C1

  >>> EXPORT MOLCAS_MAXITER=100
  >>> Do while

   &SEWARD; &SCF; Title="H2O restricted minimum"; KSDFT=B3LYP
   &SLAPAF
    Internal Coordinates
       b1 = Bond O1 H2
       b2 = Bond O1 H3
       a1 = Angle H2 O1 H3
    Vary
       a1
    Fix
       b1
       b2
    End of Internal

  >>> EndDo

In the final output, the two O--H bond lengths remain at the initial values, while the H1--O1--H2 angle is optimized
to a final angle of 112\ |o|.

The next step entails the computation of a transition state, a structure connecting different regions of
the potential energy hypersurface, and is a maximum for only one degree of
freedom. The most common saddle points have order one, that is, they are maxima for one of
one displacement and minima for the others. The simplest way to search for a
transition state in |molcas| is to add the keyword :kword:`TS` to the
:program:`SLAPAF` input. Keyword :kword:`PRFC` is suggested in order to verify
the nature of the transition structure. Searching for transition states is,
however, not an easy task. An illustration of the input required for transition state optimization for water at the DFT level
is given below:

.. extractfile:: problem_based_tutorials/Water_TS.xyz

  3
  water in Transition state in bohr
  O1             0.750000        0.000000        0.000000
  H2             1.350000        0.000000        1.550000
  H3             1.350000        0.000000       -1.550000

.. extractfile:: problem_based_tutorials/DFT.transition_state.H2O.input

  *DFT transition state optimization of the water molecule
  *File: DFT.transition_state.H2O
  *
  &Gateway
   Coord=Water_TS.xyz
   Basis=ANO-S-VDZ
   Group=C1
  >>> Do while

   &SEWARD
   &SCF; Title="H2O TS optimization"; KSDFT=B3LYP
   &SLAPAF ; ITER=20 ; TS

  >>> EndDo

The initial coordinates were chosen in units of Bohr, to illustrare that this is the
default case. The optimal geometry for ground state of water is a structure with :math:`C_{2v}` symmetry.
A transition state has been found with a linear H--O--H angle of 180\ |o|.
In many cases, there may be a clue along the energy pathway for a chemical reaction about the nature of the transition state structure,
which typically represents an intermediate conformation between reactants and products.
If this turns out to be the case, it is possible to help the optimization process
proceed toward an informed guess, by invoking the keyword :kword:`FindTS` in :program:`SLAPAF`.
:kword:`FindTS` must to be accompanied with a definition of constrained geometric definitions.
:program:`SLAPAF` will guide the optimization of the transition state towards a region in
which the restriction is fulfilled. Once there, the restriction will be released
and a free search of the transition state will be performed. This technique is
frequently quite effective and makes it possible to find difficult transition
states or reduce the number of required iterations. Here, an example is provided, in
which the initial geometry of water is clearly bent, and a trial restraint is imposed
such that the angle for the transition state should be near 180\ |o|. The
final transition state will, however, be obtained without any type of geometrical restriction.

.. extractfile:: problem_based_tutorials/DFT.transition_state_restricted.H2O.input

  *DFT transition state optimization of the water molecule with geometrical restrictions
  *File: DFT.transition_state_restricted.H2O
  *
  &Gateway
   Coord=Water_TS.xyz
   Basis=ANO-S-VDZ
   Group=C1
   Constraints
     a = Angle H2 O1 H3
   Value
     a = 180.0 degree
   End of Constraints

  >>> Do while

   &SEWARD
   &SCF; Title="H2O TS optimization"; KSDFT=B3LYP
   &SLAPAF ;FindTS

  >>> EndDo

The :program:`CASPT2` geometry optimizations are somewhat different because :program:`ALASKA`
is not suited to compute :program:`CASPT2` analytical gradients. Therefore the :program:`ALASKA`
program is automatically substituted by program :program:`NUMERICAL_GRADIENT`, which will take care
of performing numerical gradients. From the user point of view the only requirement is to place
the :program:`CASPT2` input after the :program:`RASSCF` input.
The CASSCF wave function has of course to be generated in each step before
performing CASPT2. To compute a numerical gradient can be quite time consuming,
although it is a task that can be nicely parallelized. In a double-sided
gradient algorithm like here a total of :math:`6N-12+1` CASPT2 calculations are performed
each pass of the optimization, where :math:`N` is the number of atoms.

.. extractfile:: problem_based_tutorials/CASPT2.minimum_optimization.H2O.input

  *CASPT2 minimum energy optimization for water
  *File: CASPT2.minimum_optimization.H2O
  *
  &GATEWAY
   coord=Water_distorted.xyz
   basis=ANO-S-MB
   group=C1

  >>> Do while

   &SEWARD
   &RASSCF; Title="H2O restricted minimum"; nActEl=8 0 0; Inactive=1; Ras2=6
   &CASPT2; Frozen=1
   &SLAPAF

  >>> EndDo

The use of spatial symmetry makes the calculations more efficient, although
they may again complicate the preparation of input files. We can repeat the previous :program:`CASPT2`
optimization by restricting the molecule to work in the :math:`C_{2v}` point group, which, by the way,
is the proper symmetry for water in the ground state. The :program:`GATEWAY` program (as no symmetry
has been specified) will identify and work with the highest available point group,
:math:`C_{2v}`. Here the molecule is placed with YZ as the molecular plane. By adding
keyword :kword:`Symmetry` containing as elements of symmetry the YZ (symbol X) and YX (symbol Z),
the point group is totally defined and the molecule properly generated. From that point the
calculations will be restricted to use symmetry restrictions. For instance, the molecular
orbitals will be classified in the four elements of symmetry of the group, :math:`a_1`, :math:`b_1`, :math:`b_2`,
and :math:`a_2`, and most of the programs will require to define the selection of the orbitals in
the proper order. The order of the symmetry labels is determined by :program:`SEWARD` and must
be checked before proceeding, because from that point the elements of symmetry will be known
by their order in :program:`SEWARD`: :math:`a_1`, :math:`b_1`, :math:`b_2`, and :math:`a_2`, for instance, will be
symmetries 1, 2, 3, and 4, respectively. :program:`SCF` does not require to specify the
class of orbitals and it can be used as a learning tool.

.. extractfile:: problem_based_tutorials/CASPT2.minimum_optimization_C2v.H2O.input

  *CASPT2 minimum energy optimization for water in C2v
  *File: CASPT2.minimum_optimization_C2v.H2O
  *
   &GATEWAY
  Title= H2O caspt2 minimum optimization
  Symmetry= X Z
  Basis set
  O.ANO-S...2s1p.
  O        0.000000  0.000000  0.000000 Angstrom
  End of basis
  Basis set
  H.ANO-S...1s.
  H1       0.000000  0.758602  0.504284 Angstrom
  End of basis

  >>> EXPORT MOLCAS_MAXITER=100
  >>> Do while

   &SEWARD
   &RASSCF; nActEl=8 0 0; Inactive=1 0 0 0; Ras2=3 1 2 0
   &CASPT2; Frozen=1 0 0 0
   &SLAPAF &END

  >>> EndDo

Thanks to symmetry restrictions the number of iterations within :program:`NUMERICAL_GRADIENT`
has been reduced to five instead of seven, because many of the deformations
are redundant within the :math:`C_{2v}` symmetry. Also, symmetry considerations are
important when defining geometrical restrictions
(see :numref:`Sections %s <UG:sec:definition_of_internal_coordinates>`
and :numref:`%s <TUT:sec:optim>`).
