def _csSpherical_q2x(self, Q, deriv = 0, out = None, var = None):
# Use adf routines to compute derivatives
#
q = dfun.X2adf(Q, deriv, var)
# q[0] is r, q[1] = theta, q[2] = phi
if self.angle == 'rad':
sintheta = adf.sin(q[1])
costheta = adf.cos(q[1])
sinphi = adf.sin(q[2])
cosphi = adf.cos(q[2])
else: # degree
deg2rad = np.pi/180.0
sintheta = adf.sin(deg2rad * q[1])
costheta = adf.cos(deg2rad * q[1])
sinphi = adf.sin(deg2rad * q[2])
cosphi = adf.cos(deg2rad * q[2])
r = q[0]
x = r * sintheta * cosphi
y = r * sintheta * sinphi
z = r * costheta
return dfun.adf2array([x,y,z], out)
def _csPolar_q2x(self, Q, deriv = 0, out = None, var = None):
# Use adf routines to compute derivatives
#
q = dfun.X2adf(Q, deriv, var)
# q[0] is r, q[1] = phi (in deg or rad)
if self.angle == 'rad':
x = q[0] * adf.cos(q[1])
y = q[0] * adf.sin(q[1])
else: # degree
deg2rad = np.pi/180.0
x = q[0] * adf.cos(deg2rad * q[1])
y = q[0] * adf.sin(deg2rad * q[1])
return dfun.adf2array([x,y], out)
def _ftheta(self, X, deriv = 0, out = None, var = None):
"""
basis evaluation function
Use recursion relations for associated Legendre
polynomials.
"""
nd,nvar = dfun.ndnvar(deriv, var, self.nx)
if out is None:
base_shape = X.shape[1:]
out = np.ndarray( (nd, self.nf) + base_shape, dtype = X.dtype)
# Create adf object for theta
theta = dfun.X2adf(X, deriv, var)[0]
m = abs(self.m) # |m|
sinm = adf.powi(adf.sin(theta), m)
cos = adf.cos(theta)
# Calculate Legendre polynomials
# with generic algebra
F = _leg_gen(sinm, cos, m, np.max(self.l))
# Convert adf objects to a single
# derivative array
dfun.adf2array(F, out)
return out
def calcsurf(q):
max_pow = [5, 5, 5, 6, 6,
6] # max_pow[5] is really the max freq. of dihedral
qpow = []
for i in range(5):
qi = [adf.const_like(1.0, q[i]), q[i]]
for p in range(2, max_pow[i] + 1):
qi.append(qi[1] * qi[p - 1]) # qi ** p
qpow.append(qi)
# Calculate cos(n*q6)
cosq = [adf.cos(n * q[5]) for n in range(max_pow[5] + 1)]
qpow.append(cosq)
v = 0.0
nterms = powers.shape[0]
for i in range(nterms):
c = coeffs[i]
v += c * \
qpow[0][powers[i,0]] * \
qpow[1][powers[i,1]] * \
qpow[2][powers[i,2]] * \
qpow[3][powers[i,3]] * \
qpow[4][powers[i,4]] * \
qpow[5][powers[i,5]]
return v
def _csv3_q2x(self, Q, deriv = 0, out = None, var = None):
"""
Triatomic valence coordinate system Q2X instance method.
See :meth:`CoordSys.Q2X` for details.
Parameters
----------
Q : ndarray
Shape (self.nQ, ...)
deriv, out, var :
See :meth:`CoordSys.Q2X` for details.
Returns
-------
out : ndarray
Shape (nd, self.nX, ...)
"""
natoms = 3
base_shape = Q.shape[1:]
if var is None:
var = [0, 1, 2] # Calculate derivatives for all Q
nvar = len(var)
# nd = adf.nck(deriv + nvar, min(deriv, nvar)) # The number of derivatives
nd = dfun.nderiv(deriv, nvar)
# Create adf symbols/constants for each coordinate
q = []
for i in range(self.nQ):
if i in var: # Derivatives requested for this variable
q.append(adf.sym(Q[i], var.index(i), deriv, nvar))
else: # Derivatives not requested, treat as constant
q.append(adf.const(Q[i], deriv, nvar))
# q = r1, r2, theta
if out is None:
out = np.ndarray( (nd, 3*natoms) + base_shape, dtype = Q.dtype)
out.fill(0) # Initialize out to 0
# Calculate Cartesian coordinates
if self.angle == 'deg':
q[2] = (np.pi / 180.0) * q[2]
# q[2] is now in radians
if self.supplementary:
q[2] = np.pi - q[2] # theta <-- pi - theta
if self.embedding_mode == 0:
np.copyto(out[:,2], (-q[0]).d ) # -r1
np.copyto(out[:,7], (q[1] * adf.sin(q[2])).d ) # r2 * sin(theta)
np.copyto(out[:,8], (-q[1] * adf.cos(q[2])).d ) # -r2 * cos(theta)
elif self.embedding_mode == 1:
np.copyto(out[:,0], (q[0] * adf.cos(q[2]/2)).d ) # r1 * cos(theta/2)
np.copyto(out[:,2], (q[0] * adf.sin(q[2]/2)).d ) # r1 * sin(theta/2)
np.copyto(out[:,6], (q[1] * adf.cos(q[2]/2)).d ) # r2 * cos(theta/2)
np.copyto(out[:,8], (-q[1] * adf.sin(q[2]/2)).d ) # -r2 * sin(theta/2)
else:
raise RuntimeError("Unexpected embedding_mode")
return out
def _Philm(self, X, deriv = 0, out = None, var = None):
# Setup
nd,nvar = dfun.ndnvar(deriv, var, self.nx)
if out is None:
base_shape = X.shape[1:]
out = np.ndarray( (nd, self.nf) + base_shape, dtype = X.dtype)
# Make adf objects for theta and phi
x = dfun.X2adf(X, deriv, var)
theta = x[0]
phi = x[1]
#########################################
#
# Calculate F and f factors first
#
sinth = adf.sin(theta)
costh = adf.cos(theta)
lmax = np.max(self.l)
# Calculate the associated Legendre factors
Flm = []
amuni = np.unique(abs(self.m)) # list of unique |m|
for aM in amuni:
# Order aM = |M|
sinM = adf.powi(sinth, aM)
# Calculate F^M up to l <= lmax
Flm.append(_leg_gen(sinM, costh, aM, lmax))
#Flm is now a nested list
# Calculate the phi factors
smuni = np.unique(self.m) # list of unique signed m
fm = []
for sM in smuni:
if sM == 0:
fm.append(adf.const_like(1/np.sqrt(2*np.pi), phi))
elif sM < 0:
fm.append(1/np.sqrt(np.pi) * adf.sin(abs(sM) * phi))
else: #sM > 0
fm.append(1/np.sqrt(np.pi) * adf.cos(sM * phi))
#
##############################################
###########################################
# Now calculate the list of real spherical harmonics
Phi = []
for i in range(self.nf):
l = self.l[i]
m = self.m[i] # signed m
# Gather the associated Legendre polynomial (theta factor)
aM_idx = np.where(amuni == abs(m))[0][0]
l_idx = l - abs(m)
F_i = Flm[aM_idx][l_idx]
# Gather the sine/cosine (phi factor)
sM_idx = np.where(smuni == m)[0][0]
f_i = fm[sM_idx]
# Add their product to the list of functions
Phi.append(F_i * f_i)
#
###########################################
# Convert the list to a single
# DFun derivative array
dfun.adf2array(Phi, out)
# and return
return out
def _chinell(self, X, deriv = 0, out = None, var = None):
# Setup
nd,nvar = dfun.ndnvar(deriv, var, self.nx)
if out is None:
base_shape = X.shape[1:]
out = np.ndarray( (nd, self.nf) + base_shape, dtype = X.dtype)
# Make adf objects for theta and phi
x = dfun.X2adf(X, deriv, var)
r = x[0] # Cylindrical radius
phi = x[1] if self._rad else x[1] * np.pi/180.0 # Angular coordinate
#########################################
#
# Calculate R and f factors first
#
# Calculate the radial wavefunctions
Rnell = []
abs_ell_uni = np.unique(abs(self.ell)) # list of unique |ell|
expar2 = adf.exp(-0.5 * self.alpha * (r*r)) # The exponential radial factor
for aELL in abs_ell_uni:
nmax = round(np.floor(self.vmax - aELL) / 2)
Rnell.append(_radialHO_gen(r, expar2, nmax, aELL, 2, self.alpha))
# Calculate the phi factors, f_ell(phi)
sig_ell_uni = np.unique(self.ell) # list of unique signed ell
fell = []
if self._rad: # Normalization assuming radians
for sELL in sig_ell_uni:
if sELL == 0:
fell.append(adf.const_like(1/np.sqrt(2*np.pi), phi))
elif sELL < 0:
fell.append(1/np.sqrt(np.pi) * adf.sin(abs(sELL) * phi))
else: #sELL > 0
fell.append(1/np.sqrt(np.pi) * adf.cos(sELL * phi))
else: # Normalization for degrees
for sELL in sig_ell_uni:
if sELL == 0:
fell.append(adf.const_like(1/np.sqrt(360.00), phi))
elif sELL < 0:
fell.append(1/np.sqrt(180.0) * adf.sin(abs(sELL) * phi))
else: #sELL > 0
fell.append(1/np.sqrt(180.0) * adf.cos(sELL * phi))
#
##############################################
###########################################
# Now calculate the list of real 2-D HO wavefunctions
chi = []
for i in range(self.nf):
ell = self.ell[i]
n = self.n[i]
# Gather the radial factor
abs_ELL_idx = np.where(abs_ell_uni == abs(ell))[0][0]
R_i = Rnell[abs_ELL_idx][n]
# Gather the angular factor
sig_ELL_idx = np.where(sig_ell_uni == ell)[0][0]
f_i = fell[sig_ELL_idx]
# Add their product to the list of functions
chi.append(R_i * f_i)
#
###########################################
# Convert the list to a single
# DFun derivative array
dfun.adf2array(chi, out)
# and return
return out
def _zmat_q2x(self, Q, deriv=0, out=None, var=None):
"""
ZMAT X(Q) function.
Parameters
----------
Q : ndarray
Input coordinate array (`self.nQ`, ...)
deriv : int, optional
Maximum derivative order. The default is 0.
out : ndarray, optional
Output location (nd, `self.nX`, ...) . The default is None.
var : list of int, optional
Requested ordered derivative variables. If None, all are used.
Returns
-------
out : ndarray
The X coordinates.
"""
natoms = self.natoms # Number of non-dummy atoms
base_shape = Q.shape[1:]
if var is None:
var = [i for i in range(self.nQ)]
nvar = len(var) # The number of derivative variables
nd = adf.nderiv(deriv, nvar) # The number of derivatives
# Set up output location
if out is None:
out = np.ndarray((nd, 3 * natoms) + base_shape, dtype=Q.dtype)
out.fill(0) # Initialize out to 0
# Create adf symbols/constants for each CS coordinate
q = []
for i in range(self.nQ):
if i in var: # Derivatives requested for this variable
q.append(adf.sym(Q[i], var.index(i), deriv, nvar))
else: # Derivatives not requested, treat as constant
q.append(adf.const(Q[i], deriv, nvar))
# Calculate ZMAT atomic Cartesian positions, A
#
nA = len(
self._atomLabels) # The number of ZMAT entries, including dummy
A = np.ndarray((nA, 3), dtype=adf.adarray) # ZMAT atom positions
Ci = np.ndarray((3, ), dtype=adf.adarray) # ZMAT coordinate for row i
# For each atom (dummy + non-dummy) in the ZMAT
for i in range(nA):
# First, calculate the current row's ZMAT coordinates
# C = [r, theta, tau]
for j in range(3):
if j - i >= 0:
# ZMAT coordinate not defined; will not be referenced
continue
if self._coordID[i][j] is None:
# C[i,j] is a constant literal value
Ci[j] = adf.const(np.full(base_shape, self._val[i][j]),
deriv, nvar)
else:
# A value*coordinate entry
# Use coordID[i,j] to lookup the correct CS coordinate
# adarray in q
Ci[j] = self._val[i][j] * q[self._coordID[i][j]]
# Convert angular coordinates to correct unit
if j == 1 or j == 2:
if self.angles == 'rad':
pass # already correct
elif self.angles == 'deg':
Ci[j] = Ci[j] * (np.pi / 180.0) # convert deg to rad
else:
raise ValueError("Invalid angle units")
if j == 1 and self.supplementary:
# Convert from supplementary to interior angle
Ci[j] = np.pi - Ci[j]
# Now calculate the position of atom i, A[i]
#
# There are several special cases:
# Case 1: the first atom is at the origin
if i == 0:
for j in range(3):
A[0, j] = adf.const(np.zeros(base_shape), deriv, nvar)
# Case 2: the second atom is on the +z axis
elif i == 1:
# x and y are 0
A[1, 0] = adf.const(np.zeros(base_shape), deriv, nvar) # x
A[1, 1] = adf.const(np.zeros(base_shape), deriv, nvar) # y
#
# The z-value is given by r = Ci[0]
A[1, 2] = Ci[0].copy()
# Case 3: the third atom is in the z/+x plane
elif i == 2:
# There are two possible sub-cases:
# a) References are 1, then 2
# b) References are 2, then 1
#
# In both cases, y = 0 and
# x = r * sin(theta * pi/180 )
A[2, 1] = adf.const(np.zeros(base_shape), deriv, nvar) # y
A[2, 0] = Ci[0] * adf.sin(Ci[1]) # x
#
# the z coordinate depends on the sub-case
if self._ref[2] == (1, 2, None): # Case 3a)
# 1---2 +---> z
# / |
# @ x v
#
# z = r * cos(theta * pi/180)
A[2, 2] = Ci[0] * adf.cos(Ci[1])
elif self._ref[2] == (2, 1, None): # Case 3b)
# 1---2 +---> z
# \ |
# @ x v
#
# z = z2 - r*cos(theta*pi/180)
A[2, 2] = A[1, 2] - Ci[0] * adf.cos(Ci[1])
else:
# Should not reach here, the references are invalid
raise ValueError("Invalid ZMAT reference evaluation")
# Case 4: the general case for the fourth and later atoms
else:
#
#
# This atom (A[i]) is connected sequentially
# to its references: i -> ref[0] -> ref[1] -> ref[2]
A1 = A[self._ref[i][0] - 1] #`ref` contains 1-indexed values
A2 = A[self._ref[i][1] - 1]
A3 = A[self._ref[i][2] - 1]
r21 = A2 - A1
r32 = A3 - A2
rC = np.cross(r21, r32)
# Construct the local coordinate system axes
U = r21 / adf.sqrt(r21[0] * r21[0] + r21[1] * r21[1] +
r21[2] * r21[2])
V = rC / adf.sqrt(rC[0] * rC[0] + rC[1] * rC[1] +
rC[2] * rC[2])
W = np.cross(U, V)
r = Ci[0]
th = Ci[1]
phi = Ci[2]
r_sth = r * adf.sin(th)
t1 = r * adf.cos(th) # r cos(th)
t2 = -r_sth * adf.sin(phi) # -r sin(th) sin(phi)
t3 = -r_sth * adf.cos(phi) # -r sin(th) cos(phi)
for j in range(3):
A[i, j] = A1[j] + t1 * U[j] + t2 * V[j] + t3 * W[j]
##################################################
#
# All atom positions A are now calculated
#
# Place non-dummy derivative arrays in output
k = 0
for i in range(nA):
if self._notDummy[i]: # not a dummy
# output atom k <-- ZMAT atom i
for j in range(3):
np.copyto(out[:, 3 * k + j], A[i, j].d)
k += 1
# done!
return out