Auxiliary wave function

DFT methods also use a wave function, but this wave function serves merely to obtain the electron density of the molecule, and it is from the density that the energy and all molecular properties are subsequently derived. Even though the auxiliary wave function in DFT has a single determinant form, the energy expression extracted from it incorporates static as well as dynamic electron correlation. Consequently, the DFT procedure is faster than the ab initio procedmes its time consumption scales like Hartree-Fock theory, but its accuracy is much better, and is sometimes competitive even with CASPT2. As such, DFT can treat systems of up to c. 100 or more atoms and obtain results with decent accuracy for an entire potential energy surface of an enzymatic reaction. 

The two diabatic nuclear wave functions xf and x can be expressed as linear combinations of auxiliary nuclear wave functions and, respec-... 

The best wave function of the approximate form (Eq. 11.38) may then be determined by the variational principle (Eq. II.7), either by varying the quantity p as an entity, subject to the auxiliary conditions (Eq. 11.42), or by varying the basic set fv ip2,. . ., ipN subject to the orthonormality requirement. In both ways we are lead to Hartree-Fock functions pk satisfying the eigenvalue problem... 

Using GTO bases, it cannot be expected that the variational representations of the electron waves are snfficiently accnrate far ontside the so-called molecular region , i.e. the rather limited region of space where the potential clearly deviates from the asymptotic Conlomb form. Therefore the phaseshifts of the pwc basis states cannot be obtained from the analysis of their long-range behaviour, as was done in previous works with the STOCOS bases. In the present approach, this analysis may be avoided since the K-matrix techniqne allows to determine, by equation [3] below, the phase-shift difference between the eigenfunctions of Hp and the auxiliary basis functions... 

The basic variable in density functional theory (DFT)22 is the electron density n(r). In the usual implementation of DFT, the density is calculated from the occupied single-particle wave functions (r) of an auxiliary system of noninteracting electrons... 

However, the introduction of the RI approximation led to the need for large basis sets. In old R12 method, only one single basis was used for both the electronic wave function and the RI approximation. The new formulation of R12 theory presented here uses an independent basis set denoted auxiliary basis set for the RI approximation while we employ a (much) smaller basis set for the MP2 wave function (7). This auxiliary basis set makes it possible to employ standard basis sets in explicitly correlated MP2-R12 calculations. 

Then we find that the auxiliary vector potential App - Aq = (Rmc/q)[y, -x, 0] is the origin of a current of the amplitude of the wave function, which changes the orientation of the wave function distribution. 

In this paper we present a set of ID and 2D spin-1/2 models with competing F and AF interactions for which the singlet ground-state wave function can be found exactly. This function has a special form expressed in terms of auxiliary Bose operators. This form of the wave function is similar to the MP one but with infinite matrices. For special values of model parameters it can be reduced to the standard MP form. 

One of these models is the spin- ladder with competing interactions of the ferro- and antiferromagnetic types at the F-AF transition line. The exact singlet ground-state wave function on this line is found in the special form expressed in terms of auxiliary Bose-operators. The spin correlators in the singlet state show double-spiral ordering with the period of spirals equal to the system size. 

The third and final approach to the electron correlation problem included briefly here is density functional theory (DFT), a review of which has been given by Kohn in his Nobel lecture [38], The Hohcnberg Kolin theorem [39] states that there is a one-to-one mapping between the potential V(r) in which the electrons in a molecule move, the associated electron density p(r), and the ground state wave function lP0. A consequence of this is that given the density p(r), the potential and wave function lf 0 are functionals of that density. An additional theorem provided by Kohn and Sham [40] states that it is possible to construct an auxiliary reference system of non-interacting... 

Response functions. The elementary Cooper and Peierls logarithmic divergences (19) of the interacting electron gas are also present order by order in the perturbation theory of response functions in the 2kf density-wave and superconducting channels. A scaling procedure can thus be applied in order to obtain the asymptotic properties of the real part of the retarded response functions which we will note Xm.( )- is convenient to introduce auxiliary response functions noted x ( ) [107], which are defined... 

There is also a drawback to treating electronic parameters as dynamical variables. Energy flow between the physically meaningful dynamical variables, the nuclear positions, and auxiliary dynamical variables introduced for computational reasons, the electronic wave function parameters, must be kept to a minimum. The arbitrary masses assigned to the wave function parameters as additional dynamical variables are adjusted so that the characteristic frequency of their motion is sufficiently high in comparison to nuclear... 

By requiring that the wave function single-site r-matrix is then obtained by introducing the auxiliary matrices a and b (Ebert and Gyorffy 1988 Faulkner 1977) ... 

The electronic Hamiltonian is diagonal in the rigged-BO basis and it remains now to study eq.(4). The equivalence E,(a aoi)= Ej(R aoi)= Ej(R) is fulfilled by the wave functions derived from the auxiliary model. By multiplying eq.(4) from the left by Y (p aoi) and integrate over the electronic coordinates we obtain... 

The Schrodinger wave equation and its auxiliary postulates enable us to determine certain functions k of the coordinates of a system and the time. These functions are called the Schrodinger wave functions or probability amplitude functions. The square of the absolute value of a given wave function is interpreted as a probability distribution function for the coordinates of the system in the state represented by this wave function, as will be discussed in Section lOo. The wave equation has been given this name because it is a differential equation of the second order in the coordinates of the system, somewhat similar to the wave equation of classical theory. The similarity is not close, however, and we shall not utilize the analogy in our exposition. 

For our purposes, the Schrodinger equation, the auxiliary restrictions upon the wave function % and the interpretation of the wave function are conveniently taken as fundamental postulates, with no derivation from other principles necessary. 

Inasmuch as we are going to interpret the square of the absolute value of a wave function as having the physical significance of a probability distribution function, it is not unreasonable that the wave function be required to possess certain properties, such as single-valuedness, necessary in order that this interpretation be possible and unambiguous. It has been found that a satisfactory wave mechanics can be constructed on the basis of the following auxiliary postulates regarding the nature of wave functions ... 

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