8. Effective Core Potentials

Effective core potentials (ECPs) are a useful means of replacing the core electrons in a calculation with an effective potential, thereby eliminating the need for the core basis functions, which usually require a large set of Gaussians to describe them. In addition to replacing the core, they may be used to represent relativistic effects, which are largely confined to the core. In this context, both the scalar (spin-free) relativistic effects and spin-orbit (spin-dependent) relativistic effects may be included in effective potentials. NWChem has the facility to use both, and these are described in the next two sections.

A brief recapitulation of the development of RECPs is given here, following Pacios and Christiansen^8.1. The process can be viewed as starting from an atomic Dirac-Hartree-Fock calculation, done in jj coupling, and producing relativistic effective potentials (REPs) for each $l$ and $j$ value, $U^{\rm REP}_{lj}$. From these, a local potential is extracted, which for example contains the Coulomb potential of the core electrons balanced by the part of the nuclear attraction which cancels the core electron charge. The residue is expressed in a semi-local form,

$$U^{\rm REP} = U^{\rm REP}_{LJ}(r) + \sum_{l=0}^{L-1} \sum_{j... ...}_{LJ}(r) \right] \sum_m \vert lj m \rangle \langle lj m \vert$$ (8.1)

where $L$ is one larger than the maximum angular momentum in the atom. The scalar potential is obtained by averaging the REPs for each $j$ for a given $l$ to give an averaged relativistic effective potential, or AREP,

$$U^{\rm AREP}_l(r) = \frac{1}{2l+1} \left[ lU^{\rm REP}_{l-1/2}(r) + (l+1) U^{\rm REP}_{l+1/2}(r) \right].$$ (8.2)

These are summed into the full potential.

The spin-orbit potential is obtained from the difference between the REPs for the two $j$ values for a given l, and may be represented in terms of an effective spin-orbit operator,

$$H^{\rm SO} = {\bf s} \cdot \sum_{l=1}^{L-1} \frac{2}{2l+1} ... ...le \langle lm \vert \hat l\vert lm' \rangle \langle lm' \vert.$$ (8.3)

where

$$\Delta U^{\rm REP}_{l} = U^{\rm REP}_{l+1/2}(r) - U^{\rm REP}_{l-1/2}(r).$$ (8.4)

The spin-orbit integrals generated by NWChem are the integrals over the sum, including the factor of $2/(2l+1)$, so that they may be treated as an effective spin-orbit operator without further factors introduced.

The effective potentials, both scalar and spin-orbit, are fitted to Gaussians with the form

$$r^2U_l(r) = \sum_{k} A_{lk} r^{n_{lk}} e^{-B_{lk}r^{2}}$$

where $A_{lk}$ is the contraction coefficient, $n_{lk}$ is the exponent of the ``r'' term (r-exponent), and $B_{lk}$ is the Gaussian exponent. The $n_{lk}$ is shifted by 2, in accordance with most of the ECP literature and implementations, i.e., an $n_{lk} = 0$ implies $r^{-2}$. The current implementation allows $n_{lk}$ values of only 0, 1, or 2.

8.1 Scalar ECPs

The optional directive ECP allows the user to describe an effective core potential (ECP) in terms of contracted Gaussian functions as given above. Potentials using these functions must be specified explicitly by user input in the ECP directive. This directive has essentially the same form and properties as the standard BASIS directive, except for essential differences required for ECPs. Because of this, the ECP is treated internally as a basis set. The form of the input for the ECP directive is as follows:

  ECP [<string name default "ecp basis">] \
        [print || noprint default print]

     <string tag> library [<string tag_in_lib>] \
                  <string standard_set> [file <filename>]

     <string tag> [nelec] <integer number_of_electrons_replaced>
 
        ...

     <string tag> <string shell_type>
     <real r-exponent> <real Gaussian-exponent> <real list_of_coefficients>
        ...
     
  END

ECPs are automatically segmented, even if general contractions are input. The projection operators defined in an ECP are spherical by default, so there is no need to include the CARTESIAN or SPHERICAL keyword as there is for a standard basis set. ECPs are associated with centers in geometries through tags or names of centers. These tags must match in the same manner as for basis sets the tags in a GEOMETRY and ECP directives, and are limited to sixteen (16) characters. Each center with the same tag will have the same ECP. By default, the input module prints each ECP that it encounters. The NOPRINT option can be used to disable printing. There can be only one active ECP, even though several may exist in the input deck. The ECP modules load ``ecp basis'' inputs along with any ``ao basis'' inputs present. ECPs may be used in both energy and gradient calculations.

ECPs are named in the same fashion as geometries or regular basis sets, with the default name being "ecp basis". It should be clear from the above discussion on geometries and database entries how indirection is supported. All directives that are in common with the standard Gaussian basis set input have the same function and syntax.

As for regular basis sets, ECPs may be obtained from the standard library. The names of the sets of ECPs available in the standard library (their coverage is described in Appendix A) are

• "Hay-Wadt MB (n+1) ECP"
• "Hay-Wadt VDZ (n+1) ECP"
• "LANL2DZ ECP"
• "SBKJC VDZ ECP"
• "Stuttgart RLC ECP"
• "Stuttgart RSC ECP"
• "CRENBL ECP"
• "CRENBS ECP"

The keyword nelec allows the user to specify the number of core electrons replaced by the ECP. Additional input lines define the specific coefficients and exponents. The variable <shell_type> is used to specify the components of the ECP. The keyword ul entered for <shell_type> denotes the local part of the ECP. This is equivalent to the highest angular momentum functions specified in the literature for most ECPs. The standard entries (s, p, d, etc.) for shell_type specify the angular momentum projector onto the local function. The shell type label of s indicates the ul-s projector input, p indicates the ul-p, etc.

For example, the Christiansen, Ross and Ermler ARECPs are available in the standard basis set libary named {crenbl_ecp}. To perform a calculation on uranyl (UO$_2^{2+}$) with all-electron oxygen (aug-cc-pvdz basis), and uranium with an ARECP and using the corresponding basis the following input can be used

  geometry
    U 0 0  0
    O 0 0  1.65
    O 0 0 -1.65
  end
  basis 
    U library crenbl_ecp
    O library aug-cc-pvdz
  end
  ecp
    U library crenbl_ecp
  end

The following is an example of explicit input of an ECP for H$_2$CO. It defines an ECP for the carbon and oxygen atoms in the molecule.

  ecp
    C nelec 2 # ecp replaces 2 electrons on C
    C ul      # d
            1       80.0000000       -1.60000000
            1       30.0000000       -0.40000000
            2        0.5498205       -0.03990210
   C s        # s - d 
            0        0.7374760        0.63810832
            0      135.2354832       11.00916230
            2        8.5605569       20.13797020
    C p       # p - d
            2       10.6863587       -3.24684280
            2       23.4979897        0.78505765
    O nelec 2 # ecp replaces 2 electrons on O
    O ul      # d 
            1       80.0000000       -1.60000000
            1       30.0000000       -0.40000000
            2        1.0953760       -0.06623814
    O s       # s - d
            0        0.9212952        0.39552179
            0       28.6481971        2.51654843
            2        9.3033500       17.04478500
    O p       # p - s 
            2       52.3427019       27.97790770
            2       30.7220233      -16.49630500
  end

8.2 Spin-orbit ECPs

The Spin-orbit ECPs can be used with the Density Functional Approach, but one has to run the calculations without symmetry. Note: when a Hartree-Fock method is specified the spin-orbit input will be ignored.

Spin-orbit ECPs are fitted in precisely the same functional form as the scalar RECPs and have the same properties, with the exception that there is no local potential ul, no $s$ potential and no effective charge has to be defined. Spin-orbit potentials are specified in the same way as ECPs except that the directive SO is used instead of ECP. Note that there currently are no spin-orbit ECPs defined in the standard NWChem library. The SO directive is as follows:

  SO [<string name default "so basis">] \
        [print || noprint default print]

     <string tag> <string shell_type>
     <real r-exponent> <real Gaussian-exponent> <real list_of_coefficients>
        ...
     
  END

Note: in the literature the coefficients of the spin-orbit potentials are NOT always defined in the same manner. The NWChem code assumes that the spin-orbit potential defined in the input is of the form:

$$\Delta U^{\rm NWChem}_{l} = \frac{2}{2l+1} \Delta U_{l}$$ (8.5)

For example, in the literature the Stuttgart potentials are defined as $\Delta U_{l}$ and, hence, have to be multiplied by $2/(2{l}+1)$. On the other hand, the CRENBL potentials in the published papers are defined as $\frac{l}{2l+1} \Delta U_{l}$ and, hence, have to be multiplied by $2/{l}$ (Warning: on the CRENBL website the spin-orbit potentials already have been corrected with the $2/{l}$ factor).