Computational Chemistry

This is significantly higher than with twice the energy of one beryllium atom:

2 %#14.6134355¼#29.226871;#29.2192481#(#29.226871)¼0.00762

hartree or 20.0 kJ mol#^1. If unaware that CISD is not size-consistent, one might

have thought that these widely-separated atoms are destabilized by 20 kJ mol#^1.

By comparison, the hydrogen-bonded (stabilizing) enthalpy of the water dimer

lies in the range 13–21 kJ mol#^1 (Chapter 5, reference [104]).

2. In one-determinant HF (i.e. SCF) theory, each MO has a unique energy
   (eigenvalue), but this is not so for the active MOs of a CASSCF calculation.
   Why?
   The MOs used for the active space are normally localized MOs, derived from the
   canonicalMOs (Section 5.2.3.1) by taking linear combinations of the original
   MOs of the Slater determinant. Localization has no physical consequences:C
   expressed as the “localized determinant” is in effect the same asCexpressed as
   the canonical determinant, and properties calculated from the two are identical.
   However, the canonical MOs and the localized MOs arenotthe same: in the two
   sets of MOs the coefficients of the basis functions are different, which is why
   canonical and localized MOs look different. Each canonical MO has an eigen-
   value which is approximately the negative of its ionization energy (Koopmans’
   theorem); MO coefficients and eigenvalues are corresponding columns and
   diagonal elements of theCand«matrices in Eqs. 4.60 and 5.1. Since the
   localized MOs differ mathematically from the canonical, there is no reason
   why they should have physically meaningful eigenvalues.


3. In doubtful cases, the orbitals really needed for a CASSCF calculation can
   sometimes be ascertained by examining theoccupation numbersof the active
   MOs. Look up this term for a CASSCF orbital.
   In its most general physical use,occupation numberis an integer denoting the
   number of particles that can occupy a well-defined physical state. For fermions it
   is 0 or 1, and for bosons it is any integer. This is because only zero or one
   fermion(s), such as an electron, can be in the state defined by a specified set of
   quantum numbers, while a boson, such as a photon, is not so constrained (the
   Pauli exclusion principle applies to fermions, but not to bosons). In chemistry
   the occupation number of an orbital is, in general, the number of electrons in it.
   In MO theory this can be fractional.
   In CASSCF the occupation number of the active space MO numberi(ci) is
   defined as (e.g. Cramer CJ (2004) Essentials of computational chemistry, 2nd
   edn. Wiley, Chichester, UK, p 206):

occ numb of MOi¼

XCSF

n

ðocc numbÞi;na^2 n

i.e. it is the sum, over allnconfiguration state functions (CSFs) containing MOi, of
the product of the occupation number of a CSF and the fractional contribution (a^2 )
of the CSF to the total wavefunctionC. A CSF is the same as a determinant for

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