Effective core potential

Definition

The ECP (effective core potential) represents the interaction of the core electrons with the valence electrons by a potential operator of the following form :

[math]\displaystyle{ \hat{U}(\boldsymbol{r})=U_L(r)+\sum_{l=0}^{L-1}\sum_{m=-l}^l | Y_{l,m} \rangle [U_l(r)-U_L(r)] \langle Y_{l,m} | }[/math],

where

• [math]\displaystyle{ \boldsymbol{r}=(r_\text{x},r_\text{y},r_\text{z}) }[/math]: the coordinates of the electron relative to the core center,
• [math]\displaystyle{ r=|\boldsymbol{r}| }[/math]: the distance between the electron and the core center,
• [math]\displaystyle{ L-1 }[/math]: the largest angular momentum orbital appearing in the core,
• [math]\displaystyle{ U_L(r) }[/math], [math]\displaystyle{ U_l(r) }[/math]: the radial potentials,
• [math]\displaystyle{ Y_{l,m} }[/math]: the real spherical harmonics, and
• [math]\displaystyle{ \sum_{m=-l}^l | Y_{l,m} \rangle \langle Y_{l,m} | }[/math]: the spherical harmonic projector of the angular momentum [math]\displaystyle{ l }[/math].

The potentials are normally fitted to linear combinations of Gaussians:

[math]\displaystyle{ U_L(r)-\frac{N_\text{c}}{r}=\sum_k d_{k,L}r^{n_{k,L}-2}e^{-\zeta_{k,L}r^2} }[/math],

[math]\displaystyle{ U_l(r)-U_L(r)=\sum_k d_{k,l}r^{n_{k,l}-2}e^{-\zeta_{k,l}r^2} }[/math],

where

• [math]\displaystyle{ N_\text{c} }[/math]: the number of core electrons,
• [math]\displaystyle{ d_{k,L} }[/math], [math]\displaystyle{ d_{k,l} }[/math]: the contraction coefficients,
• [math]\displaystyle{ \zeta_{k,L} }[/math], [math]\displaystyle{ \zeta_{k,l} }[/math]: the Gaussian exponents; positive real numbers, and
• [math]\displaystyle{ n_{k,L} }[/math], [math]\displaystyle{ n_{k,l} }[/math]: the powers restricted to the values [0, 1, 2].

Integrals

The ECP integral between unnormalized Cartesian GTOs is

[math]\displaystyle{ \left(\varphi(\boldsymbol{r};\zeta_\text{A},\boldsymbol{n}_\text{A},\boldsymbol{R}_\text{A})\middle|\hat{U}(\boldsymbol{r}-\boldsymbol{R}_\text{C})-\frac{N_\text{c}}{|\boldsymbol{r}-\boldsymbol{R}_\text{C}|}\middle|\varphi(\boldsymbol{r};\zeta_\text{B},\boldsymbol{n}_\text{B},\boldsymbol{R}_\text{B})\right)=\sum_k d_{k,L}\chi_k+\sum_{l=0}^{L-1} \sum_k d_{k,l} \gamma_{k,l} }[/math],

where A, B, and C denote the centers of the left-side GTO, the right-side GTO, and the potential core respectively. The integrals [math]\displaystyle{ \chi_k }[/math] and [math]\displaystyle{ \gamma_{k,l} }[/math] are expressed as below:

[math]\displaystyle{ \chi_k=\int_0^\infty dr \int d\Omega~\varphi(\boldsymbol{r};\zeta_\text{A},\boldsymbol{n}_\text{A},\boldsymbol{R}_\text{A}-\boldsymbol{R}_\text{C})r^{n_{k,L}}e^{-{\zeta_\text{C}}_{k,L}r^2} \varphi(\boldsymbol{r};\zeta_\text{B},\boldsymbol{n}_\text{B},\boldsymbol{R}_\text{B}-\boldsymbol{R}_\text{C}) }[/math],

[math]\displaystyle{ \gamma_{k,l}=\sum_{m=-l}^l \int_0^\infty dr \left(\int d\Omega~\varphi(\boldsymbol{r};\zeta_\text{A},\boldsymbol{n}_\text{A},\boldsymbol{R}_\text{A}-\boldsymbol{R}_\text{C})Y_{l,m}\right)r^{n_{k,l}}e^{-{\zeta_\text{C}}_{k,l} r^2}\left(\int d\Omega~Y_{l,m}\varphi(\boldsymbol{r};\zeta_\text{B},\boldsymbol{n}_\text{B},\boldsymbol{R}_\text{B}-\boldsymbol{R}_\text{C})\right) }[/math].

The integral [math]\displaystyle{ \chi_k }[/math] is called type 1 integral or unprojected integral, and the integral [math]\displaystyle{ \gamma_{k,l} }[/math] is called type 2 integral or projected integral.

For the details of the way to compute these integrals, see ECP integral.