Keywords: pseudo-gen

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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) rrc | (24)
Rps(r) | = | R(r) rrc | (25)

where

p(r) = c0 + c2r2 + c4r4 + c6r6 + c8r8 + c10r10 + c12r12. | (26)

On this pseudowavefunction we impose the norm conservation condition:

(Rps(r))2dr = (R(r))2dr | (27)

and continuity conditions on the wavefunction and its derivatives up to order four at the matching point:

= , n = 0,..., 4 | (28)

Continuity of the wavefunction:

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

p(rc) = log | (30)

Continuity of the first derivative of the wavefunction:

= (l + 1)rlep(r) + rl+1ep(r)p'(r) = Rps(r) + p'(r)Rps(r) | (31)

that is

p'(rc) = - . | (32)

Continuity of the second derivative of the wavefunction:

| = | (l + 1)rlep(r) + rl+1ep(r)p'(r) |
| = | l (l + 1)rl-1ep(r) +2(l + 1)rlep(r)p'(r) + rl+1ep(r)p'(r) + rl+1ep(r)p''(r) |
| = | + p'(r) + p'(r) + p''(r)rl+1ep(r). | (33)

From the radial Schrödinger equation:

= + (V(r) - )Rps(r) | (34)

that is

p''(rc) = (V(rc) - ) - 2p'(rc) - p'(rc) | (35)

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) = V'(rc) + 2p'(rc) - 2p''(rc) - 2p'(rc)p''(rc) | (36)

p''''(rc) | = | V''(rc) - 4p'(rc) + 4p''(r) |
|  | -2p'''(rc) - 2p''(rc)p''(rc) -2p'(rc)p'''(rc) | (37)

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

V(r) | = | - + | (38)
| = | 2p'(r) + p(r) + p''(r) + | (39)

Keeping only lower-order terms in r:

V(r) |  | 2(2c2r + 4c4r3) + 4c22r2 +2c2 +12c4r2 + | (40)
| = | 2c2(2l + 3) + (2l + 5)c4 + c22r2 + . | (41)

The additional constraint is:

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

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