# Vibrational normal mode calculations. # # Written by Konrad Hinsen # """ Vibrational normal modes """ __docformat__ = 'restructuredtext' from MMTK import Features, Units, ParticleProperties from Scientific import N from MMTK.NormalModes import Core # # Class for a single mode # class VibrationalMode(Core.Mode): """ Single vibrational normal mode Mode objects are created by indexing a :class:`~MMTK.NormalModes.VibrationalModes.VibrationalModes` object. They contain the atomic displacements corresponding to a single mode. In addition, the frequency corresponding to the mode is stored in the attribute "frequency". Note: the normal mode vectors are *not* mass weighted, and therefore not orthogonal to each other. """ def __init__(self, universe, n, frequency, mode): self.frequency = frequency Core.Mode.__init__(self, universe, n, mode) def __str__(self): return 'Mode ' + `self.number` + ' with frequency ' + `self.frequency` __repr__ = __str__ def effectiveMassAndForceConstant(self): m = self.massWeightedDotProduct(self)/self.dotProduct(self) k = m*(2.*N.pi*self.frequency)**2 return m, k # # Class for a full set of normal modes # class VibrationalModes(Core.NormalModes): """ Vibrational modes describe the independent vibrational motions of a harmonic system. They are obtained by diagonalizing the mass-weighted force constant matrix. In order to obtain physically reasonable normal modes, the configuration of the universe must correspond to a local minimum of the potential energy. Individual modes (see :class:`~MMTK.NormalModes.VibrationalModes.VibrationalMode`) can be extracted by indexing with an integer. Looping over the modes is possible as well. """ features = [] def __init__(self, universe=None, temperature = 300.*Units.K, subspace = None, delta = None, sparse = False): """ :param universe: the system for which the normal modes are calculated; it must have a force field which provides the second derivatives of the potential energy :type universe: :class:`~MMTK.Universe.Universe` :param temperature: the temperature for which the amplitudes of the atomic displacement vectors are calculated. A value of None can be specified to have no scaling at all. In that case the mass-weighted norm of each normal mode is one. :type temperature: float :param subspace: the basis for the subspace in which the normal modes are calculated (or, more precisely, a set of vectors spanning the subspace; it does not have to be orthogonal). This can either be a sequence of :class:`~MMTK.ParticleProperties.ParticleVector` objects or a tuple of two such sequences. In the second case, the subspace is defined by the space spanned by the second set of vectors projected on the complement of the space spanned by the first set of vectors. The first set thus defines directions that are excluded from the subspace. The default value of None indicates a standard normal mode calculation in the 3N-dimensional configuration space. :param delta: the rms step length for numerical differentiation. The default value of None indicates analytical differentiation. Numerical differentiation is available only when a subspace basis is used as well. Instead of calculating the full force constant matrix and then multiplying with the subspace basis, the subspace force constant matrix is obtained by numerical differentiation of the energy gradients along the basis vectors of the subspace. If the basis is much smaller than the full configuration space, this approach needs much less memory. :type delta: float :param sparse: a flag that indicates if a sparse representation of the force constant matrix is to be used. This is of interest when there are no long-range interactions and a subspace of smaller size then 3N is specified. In that case, the calculation will use much less memory with a sparse representation. :type sparse: bool """ if universe == None: return Features.checkFeatures(self, universe) Core.NormalModes.__init__(self, universe, subspace, delta, sparse, ['array', 'imaginary', 'frequencies', 'omega_sq']) self.temperature = temperature self.weights = N.sqrt(self.universe.masses().array) self.weights = self.weights[:, N.NewAxis] self._forceConstantMatrix() ev = self._diagonalize() self.imaginary = N.less(ev, 0.) self.omega_sq = ev self.frequencies = N.sqrt(N.fabs(ev)) / (2.*N.pi) self.sort_index = N.argsort(self.frequencies) self.array.shape = (self.nmodes, self.natoms, 3) self.cleanup() def __setstate__(self, state): # This fixes unpickled objects from pre-2.5 versions. # Their modes were stored in un-weighted coordinates. if not state.has_key('weights'): weights = N.sqrt(state['universe'].masses().array)[:, N.NewAxis] state['weights'] = weights state['array'] *= weights[N.NewAxis, :, :] if state['temperature'] is not None: factor = N.sqrt(2.*state['temperature']*Units.k_B/Units.amu) / \ (2.*N.pi) freq = state['frequencies'] for i in range(state['nmodes']): index = state['sort_index'][i] if index >= 6: state['array'][index] *= freq[index]/factor self.__dict__.update(state) def __getitem__(self, item): index = self.sort_index[item] f = self.frequencies[index] if self.imaginary[index]: f = f*1.j # !! if self.temperature is None or item < 6: amplitude = 1. else: amplitude = N.sqrt(2.*self.temperature*Units.k_B/Units.amu) / \ (2.*N.pi*f) return VibrationalMode(self.universe, item, f, amplitude*self.array[index]/self.weights) def rawMode(self, item): """ :param item: the index of a normal mode :type item: int :returns: the unscaled mode vector :rtype: :class:`~MMTK.NormalModes.VibrationalModes.VibrationalMode` """ index = self.sort_index[item] f = self.frequencies[index] if self.imaginary[index]: f = f*1.j return VibrationalMode(self.universe, item, f, self.array[index]) def fluctuations(self, first_mode=6): f = ParticleProperties.ParticleScalar(self.universe) for i in range(first_mode, self.nmodes): mode = self.rawMode(i) f += (mode*mode)/mode.frequency**2 f.array /= self.weights[:, 0]**2 s = 1./(2.*N.pi)**2 if self.temperature is not None: s *= Units.k_B*self.temperature f.array *= s return f def anisotropicFluctuations(self, first_mode=6): f = ParticleProperties.ParticleTensor(self.universe) for i in range(first_mode, self.nmodes): mode = self.rawMode(i) array = mode.array f.array += (array[:, :, N.NewAxis]*array[:, N.NewAxis, :]) \ / mode.frequency**2 s = (2.*N.pi)**2 if self.temperature is not None: s /= Units.k_B*self.temperature f.array /= s*self.universe.masses().array[:, N.NewAxis, N.NewAxis] return f def meanSquareDisplacement(self, subset=None, weights=None, time_range = (0., None, None), tau=None, first_mode = 6): """ :param subset: the subset of the universe used in the calculation (default: the whole universe) :type subset: :class:`~MMTK.Collections.GroupOfAtoms` :param weights: the weight to be given to each atom in the average (default: atomic masses) :type weights: :class:`~MMTK.ParticleProperties.ParticleScalar` :param time_range: the time values at which the mean-square displacement is evaluated, specified as a range tuple (first, last, step). The defaults are first=0, last= 20 times the longest vibration perdiod, and step defined such that 300 points are used in total. :type time_range: tuple :param tau: the relaxation time of an exponential damping factor (default: no damping) :type tau: float :param first_mode: the first mode to be taken into account for the fluctuation calculation. The default value of 6 is right for molecules in vacuum. :type first_mode: int :returns: the averaged mean-square displacement of the atoms in subset as a function of time :rtype: Scientific.Functions.Interpolation.InterpolatingFunction """ if self.temperature is None: raise ValueError("no temperature available") if subset is None: subset = self.universe if weights is None: weights = self.universe.masses() weights = weights*subset.booleanMask() total = weights.sumOverParticles() weights = weights/total first, last, step = (time_range + (None, None))[:3] if last is None: last = 20./self[first_mode].frequency if step is None: step = (last-first)/300. time = N.arange(first, last, step) if tau is None: damping = 1. else: damping = N.exp(-(time/tau)**2) msd = N.zeros(time.shape, N.Float) for i in range(first_mode, self.nmodes): mode = self[i] omega = 2.*N.pi*mode.frequency d = (weights*(mode*mode)).sumOverParticles() N.add(msd, d*(1.-damping*N.cos(omega*time)), msd) return InterpolatingFunction((time,), msd) def EISF(self, q_range = (0., 15.), subset=None, weights = None, random_vectors = 15, first_mode = 6): """ :param q_range: the range of angular wavenumber values :type q_range: tuple :param subset: the subset of the universe used in the calculation (default: the whole universe) :type subset: :class:`~MMTK.Collections.GroupOfAtoms` :param weights: the weight to be given to each atom in the average (default: incoherent scattering lengths) :type weights: :class:`~MMTK.ParticleProperties.ParticleScalar` :param random_vectors: the number of random direction vectors used in the orientational average :type random_vectors: int :param first_mode: the first mode to be taken into account for the fluctuation calculation. The default value of 6 is right for molecules in vacuum. :type first_mode: int :returns: the Elastic Incoherent Structure Factor (EISF) as a function of angular wavenumber :rtype: Scientific.Functions.Interpolation.InterpolatingFunction """ if self.temperature is None: raise ValueError("no temperature available") if subset is None: subset = self.universe if weights is None: weights = self.universe.getParticleScalar('b_incoherent') weights = weights*weights weights = weights*subset.booleanMask() total = weights.sumOverParticles() weights = weights/total first, last, step = (q_range+(None,))[:3] if step is None: step = (last-first)/50. q = N.arange(first, last, step) f = MMTK.ParticleTensor(self.universe) for i in range(first_mode, self.nmodes): mode = self[i] f = f + mode.dyadicProduct(mode) eisf = N.zeros(q.shape, N.Float) for i in range(random_vectors): v = MMTK.Random.randomDirection() for a in subset.atomList(): exp = N.exp(-v*(f[a]*v)) N.add(eisf, weights[a]*exp**(q*q), eisf) return InterpolatingFunction((q,), eisf/random_vectors) def incoherentScatteringFunction(self, q, time_range = (0., None, None), subset=None, weights=None, tau=None, random_vectors=15, first_mode = 6): """ :param q: the angular wavenumber :type q: float :param time_range: the time values at which the mean-square displacement is evaluated, specified as a range tuple (first, last, step). The defaults are first=0, last= 20 times the longest vibration perdiod, and step defined such that 300 points are used in total. :type time_range: tuple :param subset: the subset of the universe used in the calculation (default: the whole universe) :type subset: :class:`~MMTK.Collections.GroupOfAtoms` :param weights: the weight to be given to each atom in the average (default: incoherent scattering lengths) :type weights: :class:`~MMTK.ParticleProperties.ParticleScalar` :param tau: the relaxation time of an exponential damping factor (default: no damping) :type tau: float :param random_vectors: the number of random direction vectors used in the orientational average :type random_vectors: int :param first_mode: the first mode to be taken into account for the fluctuation calculation. The default value of 6 is right for molecules in vacuum. :type first_mode: int :returns: the Incoherent Scattering Function as a function of time :rtype: Scientific.Functions.Interpolation.InterpolatingFunction """ if self.temperature is None: raise ValueError("no temperature available") if subset is None: subset = self.universe if weights is None: weights = self.universe.getParticleScalar('b_incoherent') weights = weights*weights weights = weights*subset.booleanMask() total = weights.sumOverParticles() weights = weights/total first, last, step = (time_range + (None, None))[:3] if last is None: last = 20./self[first_mode].frequency if step is None: step = (last-first)/400. time = N.arange(first, last, step) if tau is None: damping = 1. else: damping = N.exp(-(time/tau)**2) finc = N.zeros(time.shape, N.Float) random_vectors = MMTK.Random.randomDirections(random_vectors) for v in random_vectors: for a in subset.atomList(): msd = 0. for i in range(first_mode, self.nmodes): mode = self[i] d = (v*mode[a])**2 omega = 2.*N.pi*mode.frequency msd = msd+d*(1.-damping*N.cos(omega*time)) N.add(finc, weights[a]*N.exp(-q*q*msd), finc) return InterpolatingFunction((time,), finc/len(random_vectors))