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Valence Angle Potentials

 

f

The valence angle and associated vectors

The valence angle potentials describe the bond bending terms between the specified atoms. They should not be confused with the three body potentials described later, which are defined by atom types rather than indices.

1. Harmonic: (harm)
   

   $$U(\theta_{jik})= {k\over 2} (\theta_{jik} - \theta_0)^2;$$ (2.12)

   
   
2. Quartic: (quar)
   

   $$U(\theta_{jik})= {k\over 2}(\theta_{jik} - \theta_0)^2 + {k... ...{jik} - \theta_0)^3 + {k''\over 4}(\theta_{jik} - \theta_0)^4;$$ (2.13)

   
   
3. Truncated harmonic: (thrm)
   

   $$U(\theta_{jik})= {k\over 2} (\theta_{jik} - \theta_0)^2 \exp[-(r_{ij}^8 + r_{ik}^8)/\rho^8];$$ (2.14)

   
   
4. Screened harmonic: (shrm)
   

   $$U(\theta_{jik})= {k\over 2} (\theta_{jik} - \theta_0)^2 \exp[-(r_{ij}/\rho_1 + r_{ik}/\rho_2)] ;$$ (2.15)

   
   
5. Screened Vessal[24]: (bvs1)
   

   $$U(\theta_{jik}) = {k \over 8(\theta_{jik}-\pi)^2}\left\{ \left[ (\theta_0 -\pi)^2 -(\theta_{jik}-\pi)^2\right]^2 \right\}$$

   $$\exp[-(r_{ij}/\rho_1 + r_{ik}/\rho_2)];$$ (2.16)

   
   

6. Truncated Vessal[25]: (bvs2 )
   

   $$U(\theta_{jik}) = k\big[ \theta_{jik}^a (\theta_{jik}-\theta_0)^2 (\theta_{jik}+\theta_0-2\pi)^2 - {a\over 2} \pi^{a-1}$$

   $$(\theta_{jik}-\theta_0)^2(\pi - \theta_0)^3\big] \exp[-(r_{ij}^8 + r_{ik}^8)/\rho^8].$$ (2.17)

   
   

7. Harmonic cosine: (hcos)
   

   $$U(\theta_{jik})={k\over 2}(cos(\theta_{jik}) -cos(\theta_{0}))^{2}$$ (2.18)

   
   
8. Cosine: (cos)
   

   $$U(\theta_{jik})=A[1+cos(m\theta_{jik}-\delta)]$$ (2.19)

   
   

In these formulae $\theta_{jik}$ is the angle between bond vectors $\mbox{$\underline{r}$}_{ij}$ and $\mbox{$\underline{r}$}_{ik}$:

$$\theta_{jik}=cos^{-1}\left\{\frac{\mbox{$\underline{r}$}_{ij}\cdot\mbox{$\underline{r}$}_{ik}} {r_{ij}r_{ik}}\right\}$$ (2.20)

In DL_POLY_2 the most general form for the valence angle potentials can be written as:

$$U(\theta_{jik},r_{ij},r_{ik})=A(\theta_{jik})S(r_{ij})S(r_{ik})$$ (2.21)

where $A(\theta)$ is a purely angular function and $S(r)$ is a screening or truncation function. All the function arguments are scalars. With this reduction the force on an atom derived from the valence angle potential is given by:

$$f_{\ell}^{\alpha}=-\frac{\partial}{\partial r_{\ell}^{\alpha}}U(\theta_{jik},r_{ij},r_{ik}),$$ (2.22)

with atomic label $\ell$ being one of $i,j,k$ and $\alpha$ indicating the $x,y,z$ component. The derivative is

$$-\frac{\partial}{\partial r_{\ell}^{\alpha}}U(\theta_{jik},r_{ij},r_{ik}) = -S(r_{ij})S(r_{ik})\frac{\partial}{\partial r_{\ell}^{\alpha}}A(\theta_{jik})$$

$$- A(\theta_{jik})S(r_{ik})(\delta_{\ell j}-\delta_{\ell i}) \frac{r_{ij}^{\alpha}}{r_{ij}} \frac{\partial}{\partial r_{ij}}S(r_{ij})$$

$$- A(\theta_{jik})S(r_{ij})(\delta_{\ell k}-\delta_{\ell i}) \frac{r_{ik}^{\alpha}}{r_{ik}} \frac{\partial}{\partial r_{ik}}S(r_{ik}),$$ (2.23)

with $\delta_{ab}=1$ if $a=b$ and $\delta_{ab}=0$ if $a\ne b$. In the absence of screening terms $S(r)$, this formula reduces to:

$$-\frac{\partial}{\partial r_{\ell}^{\alpha}}U(\theta_{jik},r... ...)= -\frac{\partial}{\partial r_{\ell}^{\alpha}}A(\theta_{jik})$$ (2.24)

The derivative of the angular function is

$$-\frac{\partial}{\partial r_{\ell}^{\alpha}}A(\theta_{jik})=... ...}_{ij}\cdot\mbox{$\underline{r}$}_{ik}}{r_{ij}r_{ik}}\right\},$$ (2.25)

with

$$\frac{\partial}{\partial r_{\ell}^{\alpha}}\left\{ \frac{\mbox{$\underline{r}$}_{ij}\cdot\mbox{$\underline{r}$}_{ik}}{r_{ij}r_{ik}}\right\} = (\delta_{\ell j}-\delta_{\ell i})\frac{r_{ik}^{\alpha}}{r_{ij}r_{ik}}+ (\delta_{\ell k}-\delta_{\ell i})\frac{r_{ij}^{\alpha}}{r_{ij}r_{ik}}-$$

$$\cos(\theta_{jik}) \left\{(\delta_{\ell j}-\delta_{\ell i})\frac{... ...}}+ (\delta_{\ell k}-\delta_{\ell i})\frac{r_{ik}^{\alpha}}{r_{ik}^{2}}\right\}$$ (2.26)

The atomic forces are then completely specified by the derivatives of the particular functions $A(\theta)$ and $S(r)$.

The contribution to be added to the atomic virial is given by

$${\cal W}=-(\mbox{$\underline{r}$}_{ij}\cdot\mbox{$\underline... ...j}+\mbox{$\underline{r}$}_{ik}\cdot\mbox{$\underline{f}$}_{k})$$ (2.27)

It is worth noting that in the absence of screening terms S(r), the virial is zero [26].

The contribution to be added to the atomic stress tensor is given by

$$\sigma^{\alpha \beta}=r_{ij}^{\alpha}f_{j}^{\beta}+ r_{ik}^{\alpha}f_{k}^{\beta}$$ (2.28)

and the stress tensor is symmetric.

In DL_POLY_2 valence forces are handled by the routine ANGFRC.

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