[Title]   Any input line beginning with '#' followed by at least one white-space character is a title line and is ignored.

[Comment]   Any text from an exclamation mark, '!', to the end of the line is regarded as comments and ignored.

[Blank line]   Any blank line, which consists of white-space characters and comment part only, is skipped by the input parser.

  The following parameters are essential, that is, they must be present in the input.

  The 'numBasis' keyword is used to specify the number of basis function used in the eigen function expansion.

numBasis number_of_basis_functions

It should be noted that the basis functions used in bx1VIBsol are the harmonic-oscillator wave functions (or Hermite-Gaussian functions), where y = x / α and α = (² / mk_f)^1/4.   N_v is a normalization constant and H_v(y) is a Hermite polynomial.   The quantum number v starts from zero (0).   The following line tells the program to use 501 basis functions, that is, v = 0 to 500.

numBasis  501

hbar2/2m (or rotConst)

  The keyword 'hbar2/2m' or its synonym 'rotConst' specifies the value of coefficient ² / 2m in the one-dimensional Hamiltonian,

hbar2/2m hbar-square_over_2m
  — or —
rotConst hbar-square_over_2I

The synonym 'rotConst' may be intuitive because, when an angle (in radian unit) is chosen as coordinate x of the motion, the coefficient becomes ² / 2I which is exactly the rotational constant of the motion.

UNIT: Unit of hbar2/2m (or rotConst) must be in cm⁻¹ U_x², where cm⁻¹ is used as the unit of energy, and U_x is the unit of coordinate assumed in the potential energy function input.

  This keyword tells the program of harmonic oscillator frequency or, exactly speaking, the energy of the vibrational quantum of the basis-set functions.

hoFreq harmonic_oscillator_quantum

This parameter is used to determine the exponent, α, of the Hermite-Gaussian basis set from the relation, Note that the value of this parameter is not deterministic, but it affects the convergence of the expansion calculation.   In other words, the program, in principle, can give enough accurate solution regardless of the choice of this parameter if sufficiently large number of basis functions are used for expansion.   In many cases, the use of the harmonic frequency calculated by quantum chemistry programs such as Gaussian is sufficient.

UNIT: Unit of hoFreq must be in cm⁻¹.

  Bending vibrations, like the umbrella inversion of ammonia (NH₃), sometimes involve motion between two minima which correspond to the exchange of equivalent nuclei.   In this case, symmetric vibrational state is allowed only for asymmetric or symmetric nuclear spin state, depending on whether the equivalent nuclei are fermions or bosons.   Vibrational partition functions must be properly calculated for this case, in most cases, by dividing the partition function by the symmetry number, two.   Though seldom at around ambient temperature, difference of the nuclear spin weight need to be properly accounted at low temperatures.   For example, for the umbrella inversion vibration of NH₂D, the nuclear spin weights are 1, 3, 1, 3, 1, ... for v = 0, 1, 2, 3, 4, ...

  In the input of bx1VIBsol, the nuclear spin statistics can be incorporated in either way, using one of the following two keywords.   These are optional input, and no weight nor symmetry number is used in the partition-function calculations.

  A keyword to set a symmetry number for partition-function calculations.

symNumbr symmetry_number

  This keyword is used to specify the periodic change of nuclear spin weights for vibrational states.

nuSpinSt length_of_period w-1 w-2 ...

The first parameter, length_of_period, specifies the length of the period and this is two for the NH₂D example shown above.   w-i's list the nuclear spin weights, and the number of weights in the list must be equal to length_of_period.   Below is an example for NH₂D.

nuSpinSt  2  1 3    ! period = 2, alternation is 1,3,1,3,...

  A keyword, 'output', can be used to control the bx1VIBsol outputs.   By default, the program creates output files containing eigen values and partition functions only.   The other output files for eigen vectors and eigen functions can be generated when specified in this input.

output name_of_output-1 [name_of_output-2]

Valid names for name_of_output-i are, 'vector', 'function', and 'all'.   The following three input examples are equivalent.

output  all
output  vector function
output  function vector

  The output of eigen vectors and eigen functions are optional because the calculation of these output needs some more computation time, especially for the calculation of eigen functions, which requires much longer computation time than that for the essential calculation, the diagonalization of the Hamiltonian matrix.

  Temperatures at which the partition functions are calculated can be specified by one of, or by any combination of, the following three types of input.

tempRange T_start T_end T_step
tempRecipRange numer start end step
tempList T1 T2 T3 ...

The first keyword 'tempRange' sets an equally spaced temperature list from T_start to T_end, with the interval T_step.   Next type of input beginning with a keyword '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 type of temperature specification starts with 'tempList'.   This input sets a list of temperatures as they listed after the keyword.   The temperature input line may appear in the input as many time as requested, for example, the follwing input is allowed.

tempList 298.15
tempRange 300 900 100
tempRecipRange 10000 5 10 1

DEFAULT: When no temperature specification is found in the input, the program uses its default which is equivalent to the following input.

tempRang 300. 1500. 100.

  The program bx1VIBsol assumes a power series potential function and an input in the following format.

potPars{
 order-1 coefficient-1
 order-2 coefficient-2
 order-3 coefficient-3
 ...
}

The order-i must be a nonnegative integer.   It should be noted that the potential function must satisfy the following condition. That is, the highest order must be an even number and the coefficient for the highest-order term must be positive.   bx1VIBsol does proceed calculation even if this condition was not satisfied since, in some occasion, the calculation may succeed with small number of basis functions and the result may be useful, though it would certainly fail when a larger number of basis functions was used to increase the accuracy.

  Table r2-1 shows a part of an example eigen value output, as viewed by a spread sheet software like Microsoft® Excel.   The first column shows the (quantum) numbers of the eigenstates, second shows the eigen values (or eigenstate energies) in cm⁻¹ unit, and the last shows the nuclear-spin statistical weights.

  The program prints the all eigen values calculated, that is, the number of eigen values are exactly the same as the numBasis input.   However, except for the case that the potential function is harmonic and the basis functions are exact solutions, the calculated eigen values for large quantum numbers are unreliable.   The program does error analysis and it determines the maximum reliable quantum number, at the threshold of 1% relative error in energy.   This threshold quantum number is printed to the console (as shown above) and in the eigen value output file as '(maximum reliable qntNbr)' which is found at the quantum number = 220 in the example shown in Table r2-1.

  First few lines of an example partition function output is shown in Table r2-2.   The first and the second column show reciprocal number of temperatures and temperatures in K unit, respectively.   The last column shows the calculated values of the partition function.

  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 r2-3 shows a part of eigen vector output.   The first row shows the eigen values exactly same as in eigen value output.   From second row, each column shows an eigen vector, a vector of coefficients for basis functions.   ci denotes a coefficients for i'th basis function.   For low eigenstates, for which the harmonic oscillator approximation is fairy good, the dominant coefficients are corresponding basis functions.

  The eigen vectors are printed in the output up to the 'maximum reliable quantum number' described above in 'Eigen vector output'.   Because of the limitation of Microsoft® Excel (maximum number of columns = 256) the output is separated in multiple CSV files when the number of eigen vectors exceeds 250.

  An example output of eigen functions is shown in Table r2-4.   Only thinned-out values around x = −0.3, −0.2, −0.1, 0, +0.1, +0.2, +0.3 are shown.   The values of the coordinate x are chosen automatically so that the major part of the eigen function at the highest quantum number is properly shown.

  Similarly to the eigen vector output, the eigen functions up to the 'maximum reliable quantum number' are printed.   Also the output is separated in multiple files when the number of eigen functions exceeds 250.