Optimization of General Coordinates

After gradients G^k have been calculated for coordinates q^k in optimization cycle k, new coordinates (or basis set exponents) q^k+1 can be obtained from the quasi-Newton update:

where F^k is the inverse of an approximate force constant matrix H^k. This method would immediately converge to the equilibrium geometry if F^k would be the inverse of the exact force constant matrix and the force field would be quadratic. In real applications usually none of these requirements is fulfilled. Often only a crude approximation to the force constant matrix H^k is known. Sometimes a unit matrix is employed (which means coordinate update along the negative gradient with all coordinates treated on an equal footing).

The optimization of nuclear coordinates in the space of internal coordinates is the default task performed by relax and does not need to be enabled. Any other optimization task requires explicit specifications in data group $optimize, which takes several possible options:

$optimize options

internal on/off

Structure optimization in internal coordinates.

redundant on/off

Structure optimization in redundant coordinates.

cartesian on/off

Structure optimization in cartesian coordinates.

basis on/off

Optimization of basis set exponents, contraction coefficients, scaling factors.

global on/off

Optimization of global scaling factor for all basis set exponents.

Note: All options except internal are switched off by default, unless they have been activated explicitly by specifying on.

Some of the options may be used simultaneously, e.g.

• internal, basis
• internal, global
• cartesian, basis

Other options have to be used exclusively, e.g.

• internal, cartesian
• basis, global

The update of the coordinates may be controlled by special options provided in data group $coordinateupdate which takes as options:

dqmax=real

Maximum total coordinate change (default: 0.3).

interpolate on/off

Calculate coordinate update by inter/extrapolation using coordinates and gradients of the last two optimization cycles (default: interpolate on) if possible.

statistics integer/off

Display optimization statistics for the integer previous optimization cycles. Without integer all available information will be displayed. off suppresses optimization statistics.

The following data blocks are used by program relax:

1. Input data from gradient programs grad, rdgrad, egrad, rimp2, mpgrad, etc.:

   $grad

   cartesian atomic coordinates and their gradients.

   $egrad

   exponents and scale factors and their gradients.

   $globgrad

   global scale factor and its gradient.

2. Input data from force constant program aoforce:

   $grad

   cartesian atomic coordinates and their gradients.

   $globgrad

   global scale factor and its gradient.

   $hessian

   the force constant matrix in the space of cartesian coordinates.

3. Output data from program relax:

   $coord

   cartesian atomic coordinates.

   $basis

   exponents and scale factors.

   $global

   global scale factor.

For structure optimizations the use of (redundant) internal coordinates is recommended, see Section 4.0.6. Normally internal coordinates are not used for input or output by the electronic structure programs (dscf, mpgrad, etc.). Instead the coordinates, gradients, etc. are automatically converted to internal coordinates by relax on input and the updated positions of the nuclei are written in cartesians coordinates to the data group $coord. Details are explained in the following sections.