Title:  Input line beginning with '#' followed by at least one white-space character is a title line and is ignored.   A title line differs from a comment line (a line beginning with '!') in bx1fitPlls input, but is no longer different in the input for solvers; bx1VIBsol bx1HRsol, and bx1S2HRsol.

Comment:  Any text from an exclamation mark, '!', to the end of the line is regarded as comments and ignored.   Any blank line, which consists of white-space characters only, is also skipped by the input parser.

  Sets the size of basis-set by the maximum J quantum number.   The basis functions used by bx1S2HRsol are the latitudinal part solutions of the two-dimensional free rotor, i.e., the associated Legendre functions, P_J^m(z), with respect to  z (= cos θ), where J = 0, 1, 2, ... and m = 0, ±1, ..., ±J.   The following example input tells the program to use basis functions up to J = 70.

maxJ      70      ! maximum J

Since the azimuthal part is independent and is the pure one-dimensional free rotor, the Hamiltonian matrix is diagonalized for each |m|, and the size of the matrix is 71 × 71 (J = 0–70) for |m| = 0, 70 × 70 (J = 0–69) for |m| = 1, 69 × 69 (J = 0–68) for |m| = 2, and so on.

  Specifies the rotational constant, B (= ² / 2I), of the rotor The unit of rotConst must be cm⁻¹, which is also used as the unit of energy.

  Tells the solver [to perform | not to perform] the precision analysis.   Default is yes (recommended).

  Sets the relative tolerance of the eigenvalues used in the precision analysis.   Default is 0.01 and must be in between 1×10⁻⁶ to 0.1.

  Similarly to the free rotation of a symmetric molecule, nuclear spin statistics need to be properly considered when evaluating the partition functions.   For example, for the two-dimensional hindered rotation of H₂ in the sample problem described in the Quick start: step-4, the rotational states symmetric to the permutation of two identical hydrogen nuclei (even-J states) occur only for the antisymmetric nuclear spin state (para-H₂) with T (total nuclear spin) = 0 and thus the statistical weight = 1, while odd-J states occur for ortho-H₂ with weight = 3.
  The key nuSpinSt can be used to specify the nuclear statistical weights.   Alternatively, as often used as a good approximation, the partition function may be calculated with the symmetric number specified with symNumbr.

  Specifies the periodic alternation of the statistical weight with J.   The first parameter l_period specifies the length of the period, followed by the list of weights.   Below is an example for the internal H₂ rotation in the sample problem 3. (see Quick start: step-4)

nuSpinSt  2  1 3  ! period = 2,  1,3,1,3,... (H2)
!nuSpinSt  2  2 1  ! period = 2,  2,1,2,1,... (D2)

The commented-out second line is for the case of D₂.

  Sets a symmetry number for partition-function calculations.

  The 'output' key can be used to control the outputs.   By default, the program creates output files containing eigen values and partition functions only.   The other output files (eigen vectors and eigen functions) are created when specified. Valid names (name_of_output-i) are vector, function, and all.   The following three examples are equivalent.

output  all
output  vector function
output  function vector

It should be noted that the calculation of eigen functions often needs much longer computation time than that required for the essential calculation, diagonalization of the Hamiltonian matrix.

  Specifies the maximum J for eigen vector and eigen function output.   Default is maxJ × 0.3.

  Temperatures at which the partition functions are calculated can be specified by any combination of the following three types of input. The key tempRange sets an equally spaced temperature list from T_start to T_end, with the interval T_step.   The next key tempRecipRange sets a list of temperatures equally spaced in its reciprocal numbers, that is, they are equally spaced in Arrhenius plots.   Like a convention used the abscissa of Arrhenius plots, the reciprocal temperature may be specified in the form like 1000 / T, 10000 / T, or etc. by setting the numer (numerator) to 1000, 10000, or etc.   For example,

tempRecipRange 10000 5 40 1

sets a list of temperatures corresponding to 10000 / T = 5, 6, 7, ..., 40.   The last ley tempList sets a list of temperatures as they listed.   The temperature input may appear as many time as requested.   For example, the following input is valid.

tempList 298.15
tempRange 300 900 100
tempRecipRange 10000 5 10 1

The default (used when no temperature specification is found in the input) is equivalent to the following input.

tempRange 300. 1500. 100.

  The program assumes a power series potential function (in the unit of cm⁻¹) with respect to  z (= cos θ) in the following format. The order-i must be a nonnegative integer.  

  A part of the eigen value output for sample problem 3 is shown in Table r4-1.   The third and fourth columns indicate total angular momentum, J, and its projection to z-axis, m_J.   The fifth column shows the nuclear spin statistical weight.   From the result of the precision analysis, the maximum reliable eigenvalue is marked.

  An example partition function output is shown in Table r4-2.   The first and the second column show reciprocal number of temperatures and temperatures in K unit, respectively.   The third column shows the calculated values of the partition function without energy shift, while the fourth column shows the partition function calculated by shifting the energy of the lowest eigenstate to be zero.

  If the estimated error in the partition function exceed the predefined threshold (1% relative error), the temperature list is truncated by removing higher temperatures so that the estimate error does not exceed the threshold at all the temperatures in the list.   When the program removed some temperatures from list, it print a warning message to the console.   The estimated error at the maximum temperature calculated is printed to the console.

  Table r4-3 shows a part of eigen vector output for m_J = 0.   Each column shows the coefficients for the basis functions in the order of the quantum number J (from m_J  to maxJ).   The eigen vectors up to outJmax are printed in the output.   The output is separated in multiple CSV files when the number of eigen vectors exceeds 250.

  The solver solves the problem with the normalized form of the Hamiltonian, and the eigen values printed in these files are normalized.   The eigen state energies (as reported in the eigen value output) can be calculated by multiplying the rotational constant, B.

  A part of eigen function output for m_J = 0 is shown in Table r4-4.   Each column shows an eigen function against z (= cos θ) for J = m_J  to outJmax.   The output is separated in multiple CSV files when the number of eigen functions exceeds 250.

  It should be noted that the eigen functions are given for the Hamiltonian of eq (4-1), i.e., given as a function of z.   In order to obtain the eigen function for the colatitude, θ, they must be properly converted.