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B. Equations for the Troullier-Martins method

We assume a pseudowavefunction Rps having the following form:

Rps(r) = rl+1ep(r)    r$\displaystyle \le$rc (24)
Rps(r) = R(r)    r$\displaystyle \ge$rc (25)

where

p(r) = c0 + c2r2 + c4r4 + c6r6 + c8r8 + c10r10 + c12r12. (26)
On this pseudowavefunction we impose the norm conservation condition:

$\displaystyle \int_{{r<r_c}}^{}$(Rps(r))2dr = $\displaystyle \int_{{r<r_c}}^{}$(R(r))2dr (27)
and continuity conditions on the wavefunction and its derivatives up to order four at the matching point:

$\displaystyle {d^nR^{ps}(r_c)\over dr^n}$ = $\displaystyle {d^nR(r_c)\over dr^n}$,    n = 0,..., 4 (28)

$ \bullet$ Continuity of the wavefunction:

Rps(rc) = rcl+1ep(rc) = R(rc) (29)

p(rc) = log$\displaystyle {R(r_c)\over r_c^{l+1}}$ (30)

$ \bullet$ Continuity of the first derivative of the wavefunction:

$\displaystyle {dR^{ps}(r)\over dr}$ = (l + 1)rlep(r) + rl+1ep(r)p'(r) = $\displaystyle {l+1\over r}$Rps(r) + p'(r)Rps(r) (31)
that is

p'(rc) = $\displaystyle {dR(r_c)\over dr}$$\displaystyle {1\over R^{ps}(r_c)}$ - $\displaystyle {l+1\over r_c}$. (32)

$ \bullet$ Continuity of the second derivative of the wavefunction:

$\displaystyle {d^2R^{ps}(r)\over d^2r}$ = $\displaystyle {d \over dr}$$\displaystyle \left(\vphantom{(l+1)r^le^{p(r)} + r^{l+1}e^{p(r)}p'(r)}\right.$(l + 1)rlep(r) + rl+1ep(r)p'(r)$\displaystyle \left.\vphantom{(l+1)r^le^{p(r)} + r^{l+1}e^{p(r)}p'(r)}\right)$  
  = l (l + 1)rl-1ep(r) +2(l + 1)rlep(r)p'(r) + rl+1ep(r)$\displaystyle \left[\vphantom{p'(r)}\right.$p'(r)$\displaystyle \left.\vphantom{p'(r)}\right]^{2}_{}$ + rl+1ep(r)p''(r)  
  = $\displaystyle \left(\vphantom{ {l(l+1)\over r^2}+ {2(l+1)\over r}p'(r) +
\left[p'(r)\right]^2 + p''(r) }\right.$$\displaystyle {l(l+1)\over r^2}$ + $\displaystyle {2(l+1)\over r}$p'(r) + $\displaystyle \left[\vphantom{p'(r)}\right.$p'(r)$\displaystyle \left.\vphantom{p'(r)}\right]^{2}_{}$ + p''(r)$\displaystyle \left.\vphantom{ {l(l+1)\over r^2}+ {2(l+1)\over r}p'(r) +
\left[p'(r)\right]^2 + p''(r) }\right)$rl+1ep(r). (33)

From the radial Schrödinger equation:

$\displaystyle {d^2R^{ps}(r)\over dr^2}$ = $\displaystyle \left(\vphantom{ {l(l+1)\over r^2} +{2m\over\hbar^2}(V(r)-\epsilon)}\right.$$\displaystyle {l(l+1)\over r^2}$ + $\displaystyle {2m\over\hbar^2}$(V(r) - $\displaystyle \epsilon$)$\displaystyle \left.\vphantom{ {l(l+1)\over r^2} +{2m\over\hbar^2}(V(r)-\epsilon)}\right)$Rps(r) (34)
that is

p''(rc) = $\displaystyle {2m\over\hbar^2}$(V(rc) - $\displaystyle \epsilon$) - 2$\displaystyle {l+1\over r_c}$p'(rc) - $\displaystyle \left[\vphantom{p'(r_c)}\right.$p'(rc)$\displaystyle \left.\vphantom{p'(r_c)}\right]^{2}_{}$ (35)

$ \bullet$ Continuity of the third and fourth derivatives of the wavefunction. This is assured if the third and fourth derivatives of p(r) are continuous. By direct derivation of the expression of p''(r):

p'''(rc) = $\displaystyle {2m\over\hbar^2}$V'(rc) + 2$\displaystyle {l+1\over r^2_c}$p'(rc) - 2$\displaystyle {l+1\over r_c}$p''(rc) - 2p'(rc)p''(rc) (36)


p''''(rc) = $\displaystyle {2m\over\hbar^2}$V''(rc) - 4$\displaystyle {l+1\over r^3_c}$p'(rc) + 4$\displaystyle {l+1\over r^2_c}$p''(r)  
      -2$\displaystyle {l+1\over r_c}$p'''(rc) - 2$\displaystyle \left[\vphantom{p''(r_c)p''(r_c)}\right.$p''(rc)p''(rc)$\displaystyle \left.\vphantom{p''(r_c)p''(r_c)}\right]^{2}_{}$ -2p'(rc)p'''(rc) (37)

The additional condition: V''(0) = 0 is imposed. The screened potential is

V(r) = $\displaystyle {\hbar^2\over 2m}$$\displaystyle \left(\vphantom{{1\over R^{ps}(r)}{d^2R^{ps}(r)\over dr^2}
- {l(l+1)\over r^2}}\right.$$\displaystyle {1\over R^{ps}(r)}$$\displaystyle {d^2R^{ps}(r)\over dr^2}$ - $\displaystyle {l(l+1)\over r^2}$$\displaystyle \left.\vphantom{{1\over R^{ps}(r)}{d^2R^{ps}(r)\over dr^2}
- {l(l+1)\over r^2}}\right)$ + $\displaystyle \epsilon$ (38)
  = $\displaystyle {\hbar^2\over 2m}$$\displaystyle \left(\vphantom{ 2{l+1\over r} p'(r)+\left[p(r)\right]^2+p''(r)
}\right.$2$\displaystyle {l+1\over r}$p'(r) + $\displaystyle \left[\vphantom{p(r)}\right.$p(r)$\displaystyle \left.\vphantom{p(r)}\right]^{2}_{}$ + p''(r)$\displaystyle \left.\vphantom{ 2{l+1\over r} p'(r)+\left[p(r)\right]^2+p''(r)
}\right)$ + $\displaystyle \epsilon$ (39)

Keeping only lower-order terms in r:

V(r) $\displaystyle \simeq$ $\displaystyle {\hbar^2\over 2m}$$\displaystyle \left(\vphantom{ 2{l+1\over r} (2c_2r + 4c_4r^3)
+ 4c_2^2r^2 + 2c_2 + 12c_4r^2}\right.$2$\displaystyle {l+1\over r}$(2c2r + 4c4r3) + 4c22r2 +2c2 +12c4r2$\displaystyle \left.\vphantom{ 2{l+1\over r} (2c_2r + 4c_4r^3)
+ 4c_2^2r^2 + 2c_2 + 12c_4r^2}\right)$ + $\displaystyle \epsilon$ (40)
  = $\displaystyle {\hbar^2\over 2m}$$\displaystyle \left(\vphantom{ 2c_2(2l+3) +
\left((2l+5)c_4+c_2^2\right) r^2}\right.$2c2(2l + 3) + $\displaystyle \left(\vphantom{(2l+5)c_4+c_2^2}\right.$(2l + 5)c4 + c22$\displaystyle \left.\vphantom{(2l+5)c_4+c_2^2}\right)$r2$\displaystyle \left.\vphantom{ 2c_2(2l+3) +
\left((2l+5)c_4+c_2^2\right) r^2}\right)$ + $\displaystyle \epsilon$. (41)

The additional constraint is:

(2l + 5)c4 + c22 = 0. (42)


next up previous contents
Next: Bibliography Up: User's Guide for LD1 Previous: A..4 Numerical solution   Contents
Filippo Spiga 2016-10-04