Kinetic Energy Operator - Part 3

The second term in the above equation can be rearranged using the commutation identity

Substituting the above equation into the kinetic energy operator gives

The commutators in the above equation can be significantly simplified using the following

Commutators involving mu_sd and other functions of mu are similar. The kinetic energy operator becomes

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Kinetic Energy Operator - Part 2

Since  does not commute with mu_de or mu, the last equation from the previous entry can be written in terms of commutators between , mu_de, and mu,

Using the commutation rules from the previous entry in the above equation gives

The second commutator in the above equation can be simplified using the commutation identity

Therefore,

The second to last term in the above equation can be expanded in the following form:

Kinetic Energy Operator - Part 1

The quantum mechanical kinetic energy operator is given by

This expression can be expanded into the following form:

where we have used the variable “s” in place of the subscript “4” for notational convenience in the previous expression. If mu_de and mu are expanded in the vibrational normal coordinates and only the first term depending on s is retained, then mu_de and mu are solely functions of s, and the following commutation properties hold:

Using these commutator rules gives

This notation can be condensed further by defining the operators

which obey the commutation rules

Using these definitions, the kinetic energy operator can be rearranged in the following form:

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Electron repulsion tail integrals as a function of the R-matrix radius for s, p, d, f, and g-type continuum orbitals. Open circles correspond to a full six-dimensional numerical integration of the tail integrals, and solid lines are the result of the analytical formulae. The analytical expressions are in excellent agreement with the multidimensional numerical integrations