/
gauss.py
139 lines (117 loc) · 4.36 KB
/
gauss.py
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# Copyright (C) 2003 CAMP
# Please see the accompanying LICENSE file for further information.
import numpy as np
from gpaw.utilities import erf
def I(R, a, b, alpha, beta):
"""Calculate integral and derivatives wrt. positions of Gaussian product.
::
/ 2 2
| a0+b0 a1+b1 a2+b2 -alpha r -beta (r-R)
value = | x y z e e dr,
|
/
Returns the tuple (value, d[value]/dx, d[value]/dy, d[value]/dz).
"""
result = np.zeros(4)
R = np.array(R)
result[0] = I1(R, a, b, alpha, beta)
a = np.array(a)
for i in range(3):
a[i] += 1
result[1 + i] = 2 * alpha * I1(R, tuple(a), b, alpha, beta)
a[i] -= 2
if a[i] >= 0:
result[1 + i] -= (a[i] + 1) * I1(R, tuple(a), b, alpha, beta)
a[i] += 1
return result
def I1(R, ap1, b, alpha, beta, m=0):
if ap1 == (0, 0, 0):
if b != (0, 0, 0):
return I1(-R, b, ap1, beta, alpha, m)
else:
f = 2 * np.sqrt(np.pi**5 / (alpha + beta)) / (alpha * beta)
if np.sometrue(R):
T = alpha * beta / (alpha + beta) * np.dot(R, R)
f1 = f * erf(T**0.5) * (np.pi / T)**0.5
if m == 0:
return 0.5 * f1
f2 = f * np.exp(-T) / T**m
if m == 1:
return 0.25 * f1 / T - 0.5 * f2
if m == 2:
return 0.375 * f1 / T**2 - 0.5 * f2 * (1.5 + T)
if m == 3:
return 0.9375 * f1 / T**3 - 0.25 * f2 * (7.5 +
T * (5 + 2 * T))
if m == 4:
return 3.28125 * f1 / T**4 - 0.125 * f2 * \
(52.5 + T * (35 + 2 * T * (7 + 2 * T)))
if m == 5:
return 14.7656 * f1 / T**5 - 0.03125 * f2 * \
(945 + T * (630 + T * (252 + T * (72 + T * 16))))
if m == 6:
return 81.2109 * f1 / T**6 - 0.015625 * f2 * \
(10395 + T *
(6930 + T *
(2772 + T * (792 + T * (176 + T * 32)))))
else:
raise NotImplementedError
return f / (1 + 2 * m)
for i in range(3):
if ap1[i] > 0:
break
a = ap1[:i] + (ap1[i] - 1,) + ap1[i + 1:]
result = beta / (alpha + beta) * R[i] * I1(R, a, b, alpha, beta, m + 1)
if a[i] > 0:
am1 = a[:i] + (a[i] - 1,) + a[i + 1:]
result += a[i] / (2 * alpha) * (I1(R, am1, b, alpha, beta, m) -
beta / (alpha + beta) *
I1(R, am1, b, alpha, beta, m + 1))
if b[i] > 0:
bm1 = b[:i] + (b[i] - 1,) + b[i + 1:]
result += b[i] / (2 * (alpha + beta)) * I1(R,
a, bm1, alpha, beta, m + 1)
return result
def test_derivatives(R, a, b, alpha, beta, i):
R = np.array(R)
a = np.array(a)
a[i] += 1
dIdRi = 2 * alpha * I1(R, tuple(a), b, alpha, beta)
a[i] -= 2
if a[i] >= 0:
dIdRi -= (a[i] + 1) * I1(R, tuple(a), b, alpha, beta)
a[i] += 1
dr = 0.001
R[i] += 0.5 * dr
dIdRi2 = I1(R, tuple(a), b, alpha, beta)
R[i] -= dr
dIdRi2 -= I1(R, tuple(a), b, alpha, beta)
dIdRi2 /= -dr
R[i] += 0.5 * dr
return dIdRi, dIdRi2
class Gauss:
"""Normalised Gauss distribution
from gauss import Gauss
width=0.4
gs = Gauss(width)
for i in range(4):
print 'Gauss(i)=',gs.get(i)
"""
def __init__(self, width=0.08):
self.dtype = float
self.set_width(width)
def get(self, x, x0=0):
return self.norm * np.exp(-((x-x0) * self.wm1)**2)
def set_width(self, width=0.08):
self.norm = 1. / width / np.sqrt(2 * np.pi)
self.wm1 = np.sqrt(.5) / width
class Lorentz:
"""Normalised Lorentz distribution"""
def __init__(self, width=0.08):
self.dtype = float
self.set_width(width)
def get(self, x, x0=0):
return self.norm / ((x-x0)**2 + self.width2)
def set_width(self, width=0.08):
self.norm = width / np.pi
self.width2 = width**2