Multi-determinant wave functions

As mentioned in Chapter 5, one can think of the expansion of an unknown MO in terms of basis functions as describing the MO function in the coordinate system of the basis functions. The multi-determinant wave function (4.1) can similarly be considered as describing the total wave function in a coordinate system of Slater determinants. The basis set determines the size of the one-electron basis (and thus limits the description of the one-electron functions, the MOs), while the number of determinants included determines the size of the many-electron basis (and thus limits the description of electron correlation). 

Exact or multi-determinant wave function (general, electronic, nuclear)... 

In this framework, we restrict ourselves to single-determinant wave functions employing one Slater determinant, but all derivations can be extended to multi-determinant wave functions (125-127). 

In order to obtain expressions for the local spin expectation values, different decomposition schemes exist. One may either partition the total spin expectation value (S2) (122,124) as suggested by Mayer or the total spin operator S2 (113,114) as proposed by Clark and Davidson. The corresponding decomposition schemes for multi-determinant wave functions may be found in Refs. (125-127). 

In principle, transition-metal clusters may best be treated with multi-determinant wave-function methods (139), but in practice due to their size often only DFT calculations are feasible and method-inherent errors have to be taken care of, e.g., the problem of spin contamination and the approximate nature of the exchange-correlation functionals available. 

The most challenging and therefore the most telling example for excitation theories is C2, whose ground state has a severe multi-determinant wave function. It is known that, to obtain quantitative results (errors < 0.1 eV), one must resort to EOM-CCSDTQ [134], Figure 2-11 compares EOM-CCSD, CCSDT, and various perturbation corrections to EOM-CCSD with FCI for three excited states of C2 [126], EOM-CCSD, which is usually highly accurate, is inadequate for the two states A and B with errors approaching 2 eV. All variants of the perturbation corrections are... 

SCF calculations with multi-determinant wave functions including double excitations from valence molecular orbitals. 

Integrals of this type appear in the energy expression of multi-determinant wave functions in the cross terms between different determinants and are included here to define the notation. 

HF theory only accounts for the average electron-electron interactions, and consequently neglects the correlation between electrons. Methods that include electron correlation require a multi-determinant wave function, since HF is the best singledeterminant wave function. Multi-determinant methods are computationally much more involved than the HF model, but can generate results that systematically approach the exact solution of the Schrodinger equation. These methods are described in Chapter 4. 

As mentioned in Chapter 5, one can think of the expansion of an unknown MO in terms of basis functions as describing the MO function in the coordinate system of the basis functions. The multi-determinant wave function (eq. (4.1)) can similarly be... 

Programs to provide MPDs now available can use single determinant wave functions from calculations produced by the Gaussian suite of programs [16] for molecules, or by Crystal-98 [17] for periodic systems. MPDs can be produced also with correlated wave functions, via a Quantum Monte Carlo program, cf. [10,15], Probabilities can be computed for multi-determinant wave functions [18], but the optimization of Q is not implemented yet. 

The relative importance of tlie different excitations may qualitatively be understood by noting tliat the doubles provide electron correlation for electron pairs, Quadruply excited determinants are important as they primarily correspond to products of double excitations. The singly excited determinants allow inclusion of multi-reference charactei in the wave function, i.e. they allow the orbitals to relax . Although the HF orbitals are optimum for the single determinant wave function, that is no longer the case when man) determinants are included. The triply excited determinants are doubly excited relative tc the singles, and can then be viewed as providing correlation for the multi-reference part of the Cl wave function. 

For a multi-Slater determinant wave function, orbitals which satisfy Eq. (3.6), and therefore Eq. (3.7), can still be defined. For these orbitals, referred to as the natural spin orbitals, the coefficients nt are not necessarily integers, but have the boundaries 0 n, 1. 

The energy of a wave function containing variational parameters, such as a Hartree-Fock (one Slater determinant) or multi-configurational (many Slater determinants) wave function. Parameters are typically the molecular orbital and configurational state coefficients, but may also be for example basis function exponents. Usually only minima are desired, although in some cases saddle points may also be of interest (excited states). 

There is an application of determinants in quantum chemistry that comes from Property 2 and Property 3. The electronic wave function of a system containing two or more electrons must change sign if the coordinates of two of the electrons are interchanged (the wave function must be antisymmetric). For example, if ri and T2 are the position vectors of two electrons and i is a multi-electron wave function, then the wave function must obey... 

It is possible to construct a Cl wave function starting with an MCSCF calculation rather than starting with a HF wave function. This starting wave function is called the reference state. These calculations are called multi-reference conhguration interaction (MRCI) calculations. There are more Cl determinants in this type of calculation than in a conventional Cl. This type of calculation can be very costly in terms of computing resources, but can give an optimal amount of correlation for some problems. 

For some systems a single determinant (SCFcalculation) is insufficient to describe the electronic wave function. For example, square cyclobutadiene and twisted ethylene require at least two configurations to describe their ground states. To allow several configurations to be used, a multi-electron configuration interaction technique has been implemented in HyperChem. 

A generic multi-determinant trial wave function can be written as... 

Since the UHF wave function is multi-determinantal in terms of R(0)HF determinants, it follows that it to some extent includes electron correlation (relative to the RHF reference). 

The Multi-configuration Self-consistent Field (MCSCF) method can be considered as a Cl where not only the coefficients in front of the determinants are optimized by the variational principle, but also the MOs used for constructing the determinants are made optimum. The MCSCF optimization is iterative just like the SCF procedure (if the multi-configuration is only one, it is simply HF). Since the number of MCSCF iterations required for achieving convergence tends to increase with the number of configurations included, the size of MCSCF wave function that can be treated is somewhat smaller than for Cl methods. 

Specifically, if T] < 0.02, the CCSD(T) metliod is expected to give results close the full Cl limit for the given basis set. If is larger than 0.02, it indicates that the reference wave function has significant multi-determinant character, and multi-reference coupled cluster should preferentially be employed. Such methods are being developedbut have not yet seen any extensive use. 

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