Source code for psi4.driver.diatomic

# Psi4: an open-source quantum chemistry software package
# Copyright (c) 2007-2019 The Psi4 Developers.
# The copyrights for code used from other parties are included in
# the corresponding files.
# This file is part of Psi4.
# Psi4 is free software; you can redistribute it and/or modify
# it under the terms of the GNU Lesser General Public License as published by
# the Free Software Foundation, version 3.
# Psi4 is distributed in the hope that it will be useful,
# but WITHOUT ANY WARRANTY; without even the implied warranty of
# GNU Lesser General Public License for more details.
# You should have received a copy of the GNU Lesser General Public License along
# with Psi4; if not, write to the Free Software Foundation, Inc.,
# 51 Franklin Street, Fifth Floor, Boston, MA 02110-1301 USA.

import numpy as np

from psi4 import core
from psi4.driver import constants
from psi4.driver.p4util.exceptions import *

def least_squares_fit_polynomial(xvals, fvals, localization_point, no_factorials=True, weighted=True, polynomial_order=4):
    """Performs and unweighted least squares fit of a polynomial, with specified order
       to an array of input function values (fvals) evaluated at given locations (xvals).
       See, particularly eqn (7) for details. """
    xpts = np.array(xvals) - localization_point
    if weighted:
        R = 1.0
        p_nu = 1
        epsilon = 1e-3
        zvals = np.square(xpts/R)
        weights = np.exp(-zvals) / (zvals**p_nu + epsilon**p_nu)
        weights = None
    fit = np.polynomial.polynomial.polyfit(xpts, fvals, polynomial_order, w=weights)
    # Remove the 1/n! coefficients
    if no_factorials:
        scalefac = 1.0
        for n in range(2,polynomial_order+1):
            scalefac *= n
            fit[n] *= scalefac
    return fit

[docs]def anharmonicity(rvals, energies, plot_fit='', mol = None): """Generates spectroscopic constants for a diatomic molecules. Fits a diatomic potential energy curve using a weighted least squares approach (c.f., particularly eqn. 7), locates the minimum energy point, and then applies second order vibrational perturbation theory to obtain spectroscopic constants. Any number of points greater than 4 may be provided, and they should bracket the minimum. The data need not be evenly spaced, and can be provided in any order. The data are weighted such that those closest to the minimum have highest impact. A dictionary with the following keys, which correspond to spectroscopic constants, is returned: :type rvals: list :param rvals: The bond lengths (in Angstrom) for which energies are provided, of length at least 5 and equal to the length of the energies array :type energies: list :param energies: The energies (Eh) computed at the bond lengths in the rvals list :type plot_fit: string :param plot_fit: A string describing where to save a plot of the harmonic and anharmonic fits, the inputted data points, re, r0 and the first few energy levels, if matplotlib is available. Set to 'screen' to generate an interactive plot on the screen instead. If a filename is provided, the image type is determined by the extension; see matplotlib for supported file types. :returns: (*dict*) Keys: "re", "r0", "we", "wexe", "nu", "ZPVE(harmonic)", "ZPVE(anharmonic)", "Be", "B0", "ae", "De" corresponding to the spectroscopic constants in cm-1 """ angstrom_to_bohr = 1.0 / constants.bohr2angstroms angstrom_to_meter = 10e-10; # Make sure the input is valid if len(rvals) != len(energies): raise ValidationError("The number of energies must match the number of distances") npoints = len(rvals) if npoints < 5: raise ValidationError("At least 5 data points must be provided to compute anharmonicity") core.print_out("\n\nPerforming a fit to %d data points\n" % npoints) # Make sure the molecule the user provided is the active one molecule = mol if mol is not None else core.get_active_molecule() molecule.update_geometry() natoms = molecule.natom() if natoms != 2: raise Exception("The current molecule must be a diatomic for this code to work!") m1 = molecule.mass(0) m2 = molecule.mass(1) # Optimize the geometry, refitting the surface around each new geometry core.print_out("\nOptimizing geometry based on current surface:\n\n"); re = np.mean(rvals) maxit = 30 thres = 1.0e-9 for i in range(maxit): derivs = least_squares_fit_polynomial(rvals,energies,localization_point=re) e,g,H = derivs[0:3] core.print_out(" E = %20.14f, x = %14.7f, grad = %20.14f\n" % (e, re, g)) if abs(g) < thres: break re -= g/H; if i == maxit-1: raise ConvergenceError("diatomic geometry optimization", maxit) core.print_out(" Final E = %20.14f, x = %14.7f, grad = %20.14f\n" % (e, re, g)); if re < min(rvals): raise Exception("Minimum energy point is outside range of points provided. Use a lower range of r values.") if re > max(rvals): raise Exception("Minimum energy point is outside range of points provided. Use a higher range of r values.") # Convert to convenient units, and compute spectroscopic constants d0,d1,d2,d3,d4 = derivs*constants.hartree2aJ core.print_out("\nEquilibrium Energy %20.14f Hartrees\n" % e) core.print_out("Gradient %20.14f\n" % g) core.print_out("Quadratic Force Constant %14.7f MDYNE/A\n" % d2) core.print_out("Cubic Force Constant %14.7f MDYNE/A**2\n" % d3) core.print_out("Quartic Force Constant %14.7f MDYNE/A**3\n" % d4) hbar = constants.h / (2.0 * np.pi) mu = ((m1*m2)/(m1+m2))*constants.amu2kg we = 5.3088375e-11 * np.sqrt(d2/mu) wexe = (1.2415491e-6)*(we/d2)**2 * ((5.0*d3*d3)/(3.0*d2)-d4) # Rotational constant: Be I = ((m1*m2)/(m1+m2)) * constants.amu2kg * (re * angstrom_to_meter)**2 B = constants.h / (8.0 * np.pi**2 * constants.c * I) # alpha_e and quartic centrifugal distortion constant ae = -(6.0 * B**2 / we) * ((1.05052209e-3*we*d3)/(np.sqrt(B * d2**3))+1.0) de = 4.0*B**3 / we**2 # B0 and r0 (plus re check using Be) B0 = B - ae / 2.0 r0 = np.sqrt(constants.h / (8.0 * np.pi**2 * mu * constants.c * B0)) recheck = np.sqrt(constants.h / (8.0 * np.pi**2 * mu * constants.c * B)) r0 /= angstrom_to_meter; recheck /= angstrom_to_meter; # Fundamental frequency nu nu = we - 2.0 * wexe; zpve_nu = 0.5 * we - 0.25 * wexe; # Generate pretty pictures, if requested if(plot_fit): try: import matplotlib.pyplot as plt except ImportError: msg = "\n\tPlot not generated; matplotlib is not installed on this machine.\n\n" print(msg) core.print_out(msg) # Correct the derivatives for the missing factorial prefactors dvals = np.zeros(5) dvals[0:5] = derivs[0:5] dvals[2] /= 2 dvals[3] /= 6 dvals[4] /= 24 # Default plot range, before considering energy levels minE = np.min(energies) maxE = np.max(energies) minR = np.min(rvals) maxR = np.max(rvals) # Plot vibrational energy levels we_au = we / constants.hartree2wavenumbers wexe_au = wexe / constants.hartree2wavenumbers coefs2 = [ dvals[2], dvals[1], dvals[0] ] coefs4 = [ dvals[4], dvals[3], dvals[2], dvals[1], dvals[0] ] for n in range(3): Eharm = we_au*(n+0.5) Evpt2 = Eharm - wexe_au*(n+0.5)**2 coefs2[-1] = -Eharm coefs4[-1] = -Evpt2 roots2 = np.roots(coefs2) roots4 = np.roots(coefs4) xvals2 = roots2 + re xvals4 = np.choose(np.where(np.isreal(roots4)), roots4)[0].real + re Eharm += dvals[0] Evpt2 += dvals[0] plt.plot(xvals2, [Eharm, Eharm], 'b', linewidth=1) plt.plot(xvals4, [Evpt2, Evpt2], 'g', linewidth=1) maxE = Eharm maxR = np.max([xvals2,xvals4]) minR = np.min([xvals2,xvals4]) # Find ranges for the plot dE = maxE - minE minE -= 0.2*dE maxE += 0.4*dE dR = maxR - minR minR -= 0.2*dR maxR += 0.2*dR # Generate the fitted PES xpts = np.linspace(minR, maxR, 1000) xrel = xpts - re xpows = xrel[:, None] ** range(5) fit2 = np.einsum('xd,d', xpows[:,0:3], dvals[0:3]) fit4 = np.einsum('xd,d', xpows, dvals) # Make / display the plot plt.plot(xpts, fit2, 'b', linewidth=2.5, label='Harmonic (quadratic) fit') plt.plot(xpts, fit4, 'g', linewidth=2.5, label='Anharmonic (quartic) fit') plt.plot([re, re], [minE, maxE], 'b--', linewidth=0.5) plt.plot([r0, r0], [minE, maxE], 'g--', linewidth=0.5) plt.scatter(rvals, energies, c='Black', linewidth=3, label='Input Data') plt.legend() plt.xlabel('Bond length (Angstroms)') plt.ylabel('Energy (Eh)') plt.xlim(minR, maxR) plt.ylim(minE, maxE) if plot_fit == 'screen': else: plt.savefig(plot_fit) core.print_out("\n\tPES fit saved to %s.\n\n" % plot_fit) core.print_out("\nre = %10.6f A check: %10.6f\n" % (re, recheck)) core.print_out("r0 = %10.6f A\n" % r0) core.print_out("we = %10.4f cm-1\n" % we) core.print_out("wexe = %10.4f cm-1\n" % wexe) core.print_out("nu = %10.4f cm-1\n" % nu) core.print_out("ZPVE(nu) = %10.4f cm-1\n" % zpve_nu) core.print_out("Be = %10.4f cm-1\n" % B) core.print_out("B0 = %10.4f cm-1\n" % B0) core.print_out("ae = %10.4f cm-1\n" % ae) core.print_out("De = %10.7f cm-1\n" % de) results = { "re" : re, "r0" : r0, "we" : we, "wexe" : wexe, "nu" : nu, "ZPVE(harmonic)" : zpve_nu, "ZPVE(anharmonic)" : zpve_nu, "Be" : B, "B0" : B0, "ae" : ae, "De" : de } return results