BSPRVSE

Use a basis of B-splines to solve the (ro)vibrational Schrödinger equation, $$H{j}(R) \psi\nu(R) \equiv \left[ -\frac{1}{2 μ} \frac{d^2}{dR^2} + V(R) + \frac{j(j+1)}{2 μ R^2} \right] \psi\nu(R) = E\nu \psi\nu(R),$$ where $R$ is the internuclear distance, $\mu$ is the reduced mass of the molecule, $j$ is the rotational quantum number of the molecule, and $\nu$ is the vibrational quantum number of the molecule. The potential $V(R)$ can be purely real, or can have a purely imaginary potential added to the tail end, resulting in a complex absorbing potential (CAP). Using a purely real potential will result in calculating real-valued bound-state wavefunctions with real energies $E\nu$, although the output variables are actually complex. Using a CAP, the wavefunctions and energies become complex (the Hamiltonian is no longer Hermitian, so the eigenenergies no longer have to be real), although the complex part of $E_\nu$ for bound states should be negligible. The available imaginary potentials are as follows :

| type | form | | -------------|---------------------------------------------| | exponential | $V_\text{CAP}(R) = V(R) - i A N e^{-2 / x}$ | | linear | $V_\text{CAP}(R) = V(R) - i A x$ | | quadratic | $V_\text{CAP}(R) = V(R) - i3Ax^2/2$ | | cubic | $V_\text{CAP}(R) = V(R) - i 2A x^3$ | | quartic | $V_\text{CAP}(R) = V(R) - i 5A x^4/2$ |

where $x = (R - R0) / L$, $L$ is the length of the imaginary potential, and $R0$ is the starting value of the imaginary potential. The imaginary potential starts at $R_0$ and increases monotonically until the largest available value of $R$ given by the internuclear potential. See [1] for more details on absorbing potentials as they're implemented by this code.

Dependencies

• A working installation of LAPACK
• [optional] The Fortran Package Manager (fpm)

Building with fpm

In the package directory, just run

$ fpm build --profile release

The archive file libbsprvse.a and .mod files will be placed in the generated build subdirectory. These files are necessary to compile another program that uses this library.

Building without fpm

In the package directory, just run the compile script

$ ./compile

The archive file libbsprvse.a will be placed in the lib subdirectory and is necessary to link against when compiling another program that uses this library. The bsprvse executable will be placed in the bin subdirectory.

Testing

Test suite not yet implemented...

Using bsprvse

The bsprvse executable

Execution of the program by calling the bsprvse executable is controlled by namelist variables read via stdin, which can be supplied via stdin redirection, i.e. :

$ ./bin/bsprvse < input/TEMPLATE.namelist

and with fpm,

$ fpm run < input/TEMPLATE.namelist

The input parameters are defined in the $input_parameters namelist, which can be found in the main program at main/main.f. An example input is provided in input/TEMPLATE.namelist :

``` $input_parameters

j = 0 !! The rotational quantum number $j$

nwf = 65 !! The number of wavefunctions to calculate, from ν = 0 to ν = nwf - 1 nRwf = 1000 !! The number of $R$-values for the wavefunctions reducedmass = 7.3545994295895865 !! The reduced mass of the molecule, in atomic mass units, i.e., the mass of !! carbon-12 is 12 atomic mass units.

np = 400 !! The number of B-spline intervals order = 5 !! The order of the B-splines legpoints = 10 !! The number of Gauss-Legendre quadrature points used to calculate integrals. The available number of points is given !! in src/bsprvse_quadratureweights.f : [ 6, 8, 10, 12, 14, 16, 24, 28, 32, 36, 40, 44, 48, 64, 80 ]

R_max = 10.0 !! The largest value to consider for the internuclear potential

CAPexists = .true. !! Use a comlex absorbing potential (CAP) ? !! yes -> CAPexists = .true. !! no -> CAPexists = .false. CAPtype = 0 !! The CAP type: !! 0: exponential !! 1: linear !! 2: quadratic !! 3: cubic !! 4: quartic CAPlength = 4.0 !! The length of the CAP, in atomic units. The CAP starts at Rmax - CAPlength !! and continues to Rmax, where it takes its maximal value. Rmax is the !! largest value at which the internuclear potential was calcualted CAPstrength = 0.01 !! The CAP_strength $A$, in atomic units.

potentialfile = "data/CFppot_30.dat" !! The location for the input internuclear potential in the format !! R1 V1 !! R2 V2 !! R3 V3 !! . . !! . . !! . . !! !! where R and V are both in atomic units

output_directory = "output" !! The directory in which to write the wavefunctions and energies. !! This directory must already exist

leftBCzero = .false. !! Apply a zero boundary condition on the left ? rightBCzero = .false. !! Apply a zero boundary condition on the right ?

/ ```

bsprvse in your fpm project

To use this project within your fpm project, add the following to your fpm.toml file:

[dependencies]
bsprvse = { git = "https://github.com/banana-bred/bsprvse" }

The module bsprvse contains the public interface solve_RVSE() :

which may be invoked as solve_RVSE(R_vals, V_vals, j, reduced_mass, nwf, nR_wf, R_wf, wf, wf_nrg & , B_rot, np, legpoints, order, left_bc_zero, right_bc_zero & ) or

solve_RVSE(R_vals, V_vals, j, reduced_mass, nwf, nR_wf, R_wf, wf, wf_nrg & , B_rot, np, legpoints, order & , CAP_exists, CAP_length, CAP_type, CAP_strength & , left_bc_zero, right_bc_zero )

The input variables are as follows :

| variable | type | explanation | | ---------------- | ----------------------------- | ----------------------------------------------- | | R_vals(:) | real(dp), intent(in) | Array containing the values of the internuclear distance in atomic units| | V_vals(:) | real(dp), intent(in) | Array containing the values of the intermolecular potential in atomic units| | j | integer, intent(in) | The value $j$ in the Schrödinger equation| | reduced_mass | real(dp), intent(in) | The system's reduced mass in atomic units (electron mass = 1; not atomic mass units)| | nwf | integer, intent(in) | The number of wavefunctions to calculate| | nR_wf | integer, intent(in) | The number of $R$-grid points on which to evaluate the wavefunctions| | R_wf(:) | real(dp), intent(in) | The $R$-grid on which wavefunctions will be calculated | wf(:,:) | complex(dp), intent(out) | Array containing the values of the wavefunctions indexed as `(iR, iv)`, where `iR` runs over the| | | | internuclear distances and iv runs over the vibrational quantum number ν| | wf_nrg(:) | complex(dp), intent(out) | The energies of the wavefunctions (in atomic units)| | B_rot(:) | complex(dp), intent(out) | The rotational constant for each vibrational state in atomic units| | np | integer, intent(in) | The number of B-spline intervals| | legpoints | integer, intent(in) | The number of Gauss-Legendre quadrature points used to calculate integrals. | | | Available legpoints : `6, 8, 10, 12, 14, 16, 24, 28, 32, 36, 40, 44, 48, 64, 80` | | order | integer, intent(in) | The order of the B-splines| | CAP_exists | logical, intent(in) | Add an imaginary potential to the real internuclear potential ?| | | | yes -> `.true.` | | | | no -> `.false.` | | CAP_length | real(dp), intent(in) | The length of the imaginary potential in atomic units. Has no effect if `CAP_exists` is `false`| | CAP_strength | real(dp), intent(in) | The strength ($A$) of the CAP in atomic units| | CAP_type | integer, intent(in) | The type of CAP : | | | | 0 : exponential | | | | 1 : linear | | | | 2 : quadratic | | | | 3 : cubic | | | | 4 : quartic | | left_BC_zero | integer, intent(in) | Apply a zero condition on the left boundary ?| | right_BC_zero | integer, intent(in) | Apply a zero condition on the right boundary ?|

where dp represents double precision, as defined by real64 in the iso_fortran_env intrinsic module. The former call (without the CAP variables) is essentially the same as calling the latter call where CAP_exists is .false..

Reference(s)

[1] Á. Vibók and G. G. Balint-Kurti Parameterization of Complex Absorbing Potentials for Time-Dependent Quantum Dynamics, J. Phys. Chem. 1992, 96, 8712-8719 URL: https://pubs.acs.org/doi/pdf/10.1021/j100201a012