(sec:essentialelements.numericalgradients)=
# Numerical Gradients

If the analytic gradient is not available, the numerical gradient can
simply be requested via:

```orca
! NumGrad
```

as in the following example:

```{literalinclude} ../../orca_working_input/C05S02_121.inp
:language: orca
```

NOTE

-   Be aware that the numerical gradient is quite expensive. The time
    for one gradient calculation is equal to 6 $\times$ (number of
    atoms) $\times$ (time for one single point calculation).

-   The numerical gradient can be calculated in a multi-process run,
    using a maximum of three times the number of atoms (see section
    {ref}`sec:essentialelements.parallel.calling`).

More details on various options, geometry convergence criteria and the
like are found in section
{ref}`sec:structurereactivity.optimization`.

(sec:essentialelements.numericalgradients.optimization.notes)=
## Some Notes and Tricks

:::{note}

-   `TightSCF` in the SCF part is set as default to avoid the buildup of
    too much numerical noise in the gradients.

-   Even if the optimization does not converge, the ORCA output may
    still end with \"\*\*\*\*ORCA TERMINATED NORMALLY\*\*\*\*\".
    Therefore do not rely on the presence of this line as an indicator
    of whether the geometry optimization is converged! Rather, one
    should instead rely on the fact that, an optimization job that
    terminates because the maximum number of iterations has been
    reached, will generate the following output message:
:::

:::{Warning}
       The optimization did not converge but reached the maximum number of
       optimization cycles. Please check your results very carefully.
:::

While a successfully converged job will generate the following message
instead:

```
***********************HURRAY********************
***        THE OPTIMIZATION HAS CONVERGED     ***
*************************************************
```

:::{tip}

-   In rare cases the redundant internal coordinate optimization fails.
    In this case, you may try to use `COPT` (optimization in Cartesian
    coordinates). This will likely take many more steps to converge but
    should be stable.

-   For optimizations in Cartesian coordinates the initial guess Hessian
    is constructed in internal coordinates and thus these optimizations
    should converge only slightly slower than those in internal
    coordinates. Nevertheless, if you observe a slow convergence
    behaviour, it may be a good idea to compute a Hessian initially
    (perhaps at a lower level of theory) and use `InHess read` in order
    to improve convergence.

-   At the beginning of a TS optimization more information on the
    curvature of the PES is needed than a model Hessian can give. The
    best choice is analytic Hessian, available for HF, DFT and MP2. In
    other cases (e.g. CAS-SCF), the numerical evaluation is necessary.
    Nevertheless you do not need to calculate the full Hessian when
    starting such a calculation. With ORCA we have good experience with
    approximations to the exact Hessian. Here it is recommended to
    either directly combine the TS optimization with the results of a
    relaxed surface scan or to use the Hybrid Hessian as the initial
    Hessian, depending on the nature of the TS mode. Note that these
    approximate Hessians do never replace the exact Hessian at the end
    of the optimization, which is always needed to verify the minimum or
    first order saddle point nature of the obtained structure.
:::

(sec:essentialelements.numericalgradients.numgrads)=
## Numerical Gradients

If you want to use numerical instead of analytic gradients you have to
use

```orca
! NumGrad
```

in your input file. Additionally the settings for the numerical
differentiation can be changed:

```orca
%numgrad
  CentralDiff true # You should use two-sided numerical differentiation, but it 
                   # is possible to switch to one-sided numerical differentiation.
  DX 0.005         # Increment in Bohr for the differentiation.
  TransInvar  true # Take advantage of translation invariance
 end
```