# *LOCALIZATION¶

Performs localization of molecular orbitals localization using the Pipek-Mezey criterion . Since localized orbitals in general do not span single irreps, this module only works with $$C_1$$ symmetry.

The implementation uses an exponential parametrization combined with second-order minimization using the Newton trust-region method (see Sections 10.8 and 12.3 of for general principles.

In short, we have some function $$T(\mathbf{\kappa})$$ that we seek to minimize with respect to variational parameters $$\mathbf{\kappa}$$, where $$T^{[0]}=T(\mathbf{0})$$ is our current expansion point. In iteration $$n$$ we consider a second-order expansion of the target function

$Q_n(\mathbf{\kappa}_n) = T_n^{[0]} + \mathbf{T}_n^{[1]T}\mathbf{\kappa}_n + \frac{1}{2}\mathbf{\kappa}_n^T T_n^{[2]}\mathbf{\kappa}_n$

To the extent this expansion is valid, the (Newton) step to be taken is given by the linear equation

$T_n^{[2]}\mathbf{\kappa}_n=-\mathbf{T}_n^{[1]}$

We assume the second-order expansion to be valid within a trust radius $$h$$. If the proposed Newton step goes outside this radius, we will instead take a step onto the border of our trust region in an optimal direction. To find it, we set up a Lagrangian

$L(\mathbf{\kappa},\mu)=Q(\mathbf{\kappa}) -\frac{1}{2}\mu\left(\Vert\mathbf{\kappa}\Vert^2-h^2\right)$

where minimization in iteration $n$ now gives

$\left(T_n^{[2]}-\lambda_n I\right)\mathbf{\kappa}_n=-\mathbf{T}_n^{[1]}$

An important quantity in the algorithm is the ration between the observed and predicted changes in the target function

$r = \frac{T_{n+1}^{[0]}-T_{n}^{[0]}}{Q_{n}-T_{n}^{[0]}} = \frac{\mbox{DIFFG}}{\mbox{DIFFQ}}$

In the following scheme for the update of the trust radius is proposed (we quote):

1. If $$r>0.75$$ , increase the trust radius $$h_{n+1}=1.2h_{n}$$.

2. If $$0.25 < r <0.75$$, do not change the trust radius $$h_{n+1}=h_{n}$$.

3. If $$r <0.25$$ , reduce the trust radius $$h_{n+1}=0.7h_{n}$$.

4. If $$r<0$$ , reject the step $$\mathbf{\kappa}_n$$ and generate a new step of trust radius $$h_n\rightarrow 0.7h_n$$.

## .PRJLOC¶

Localization from projection analysis instead of Mulliken analysis. [Default: False]:

.PRJLOC


## .OWNBAS¶

Calculate fragments in their own basis. For now it is the only option which can be used with projection analysis option (.PRJLOC) [Default: False]:

.OWNBAS


## .VECREF¶

First give number of fragments to project onto. Then for each fragment give filename of MO coefficients and string specifying which orbitals will be used in the projection (Specification of orbital strings):

.VECREF
3
DFO1AT
1..6
DFH1AT
1
DFH2AT
1


## .HESLOC¶

Define model of the Hessian [Default: FULL].

Use full Hessian:

.HESLOC
FULL


Use diagonal approximation of the Hessian. If using this approximation, there is danger of converging to the stationary point instead of minima:

.HESLOC
DIAG


First the diagonal approximation of the Hessian will be used and after reaching convergence criterion the localization will switch to the construction of the full Hessian:

.HESLOC
COMB


## .CHECK¶

This keyword can be used only in combination with .HESLOC = COMB. The full Hessian will be calculated only at the last iteration. This way one can check if the procedure converged to the minimum.

## .THFULL¶

Convergence threshold for the localization process. It applies on the functional value of the localization criteria when full Hessian is calculated. It is used in .HESLOC = FULL scheme and in the second part of the .HESLOC = COMB scheme. When not specified gradient criterion or criterion of the number of negative eigenvalues equals zero will be used:

.THFULL
1.0D-13


Convergence threshold for the localization process. It applies on the gradient of the localization functional. It is used in .HESLOC = FULL or DIAG scheme and in the second part of the .HESLOC = COMB scheme. When not specified the functional value criterion or criterion of the number of negative eigenvalues equals zero will be used:

.THGRAD
1.0D-17


## .THDIAG¶

Convergence threshold for the localization process. It applies on the functional value of the localization criteria when diagonal Hessian is calculated. It can be used with the .HESLOC = DIAG keyword. If .HESLOC = COMB it is used to switch from the first to the second stage of the convergence process:

.THDIAG
1.0D-05


## .SELECT¶

Select subset of MOs for localization (use Specification of orbital strings syntax). Currently only Pipek-Mezey localization is implemented, therefore localize only occupied orbitals. Localization of virtual orbitals is not recommended.

.SELECT
1..5


## .MAXITR¶

Maximum number of iterations. Default:

.MAXITR
100


## .PRINT¶

Set print level. Default:

.PRINT
1