Reference Geometry

Near the equilibrium geometry, dependence of the energy on geometric displacements is approximately quadratic. As a result, small errors in the reference geometry will insignificantly affect computed energies, but more substantial errors (say, several hundredths of an A in covalent bond lengths) will compromise the reliability of a thermochemical calculation. 

For W2 theory, we opted for CCSD(T)/VQZ+1 as the level of theory for reference geometries. For geometries, the VQZ basis set is known to be close to the one-particle basis set limit [17, 18], while the addition of the inner polarization functions again takes care of inner polarization effects. 

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Some more recent software uses the tensor LEED approximation of Rous and Pen-dry which can save a substantial amount of computer time [2.268-2.270]. In tensor LEED the amplitudes (0) of all escaping electron waves (spots) are first calculated conventionally as described above for a certain reference geometry. Then the derivatives of these amplitudes 5Ag/5ri with respect to small displacements of each atom i in this reference geometry are calculated. These derivatives are the constituents of the "tensor". The wave amplitude for a modified model geometry where atom i is displaced by the vector Aq is then approximately given by ... 

The geometry is reoptimized at the MP2/6-31G(d) level, which is used as the reference geometry. 

A force field describes the restoring forces which occur in a molecule when the geometry of minima potential energy is disturbed. Force fields of the kind dealt with in this article describe the occurring forces with respect to a so-called strain-free molecular geometry or, more generally, with respect to a molecular reference geometry . 

The reason was simple in this case the reference geometry of C=CCT obviously cor-... 

The more cost-effective W1 theory and the more rigorous W2 theory have a lot of points in common. Aside from issues relating to the reference geometry and the zero-point energy, the main difference concerns the basis sets used in the extrapolation steps for the SCF and the valence correlation contribution. 

Some representative results can again be found in Table 2.2. At the W1 level, the G2-1 ionization potentials are reproduced with a MAD of only 0.013 eV [26], No further improvement is seen at the W2 level for this property. Note that if the B3LYP/VTZ geometry for CH4" is employed, a serious error is seen for IP(CH4) which disappears when a CCSD(T)/VTZ reference geometry is used instead. (Only BH HLYP... 

Reference geometry B3LYP/cc-pVTZ CCSD(T)/cc-pVQZ... 

Consider expanding the energy of a molecule in displacements from some reference geometry. We denote these displacements by 7o, where a runs over nonredundant displacements (3 N — 6 for a nonlinear reference geometry and 3 N — 5 for the linear case). The energy expansion is thus... 

We consider a molecular system at a reference geometry X0 in state X0) with total energy... 

First X = X0 is chosen as the reference geometry and a set of orthonormal MOs are determined as linear combinations of AOs ... 

We may now expand the Hamiltonian in geometrical distortions about the reference geometry... 

We assume that an MCSCF calculation has been carried out at the reference geometry X0 and denote the MCSCF state at this geometry by MC>. The MCSCF state may be expressed as... 

We assume that we have carried out a Hartree-Fock calculation at the reference geometry X0 and used the orbitals to determine a coupled-cluster wave function... 

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