Particle number representation

Let us turn now to the problem of creating more than two particles consider a many-electron one-determinantal wave function in its second quantized representation. As is easily seen from the previous example for the two-electron case, a many-electron one-determinantal wave function is constructed by successive application of creation operators on the vacuum state. It will be useful though to consider this problem from a somewhat different point of view. 

First of all, the short-hand notations k and ik used above in Eqs. (2.4) and (2.7) merit some discussion. Consider an abstract wave function, a ket, for example, of the following form  

This wave function can describe a state (configuration) containing at most M electrons. The quantities nj are simply occupation numbers, that is, nj is equal to zero if orbital i is empty, while it is equal to 1 if there is an electron on the i-th spinorbital. 

This way of specifying one-determinantal wave functions is called the particle number representation. The state where every occupation number is zero contains no particles, so it is just the vacuum state  

The wave function denoted previously by k in Eq. (3.4) can be written in the particle number representation as  

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The particle number representation is conceptually very important because, strictly speaking, the abstract wave functions given in this representation serve as the carrier space of the second quantized creation operators. In other words, the creation operators act on the particle-number represented wave functions. 

Put down the second quantized forms and the particle number representations of the following wave functions ... 

Consider an N-electron one-determinantal wave function V in the particle number representation ... 

We emphasize again that the symbol o does not mean equality in the mathematical sense because of the different Hilbert spaces considered. The wave function O on the left-hand side of Eq. (2.53) is represented in the L2 function space (or, in the I2 space in the case of a finite basis), while the second quantized wave function on the right-hand side of Eq. (2.53) makes use of the particle number representation. In a given basis, however, there is a one-to-one correspondence between the two representations. This permits one to apply the above correspondences [Eqs. (2.52)-(2.54)] to rewrite any first quantized wave function to the second quantized language or vice versa. 

If the creation and annihilation operators anticommute properly, they change the occupation numbers in an abstract Hilbert space of particle number representation which can be considered the same even if the physical orbitals do move (change). For this reason the true fermion operators need not be varied either. Their algebraic properties are determined by the relevant commutation rules, which are also independent of the physical properties of the system or of the nature of the basis orbitals. These anticommutation properties of the operators are the same after and before the variation. 

Again to save space, some other related methods such as Green s function formalism or the diagrammatic perturbation theory, which are usually treated with second quantization on an equal footing, are not presented here. Merely the second quantized approach (particle number representation) will be elaborately discussed. However, a short review of some recent developments partly connected to the author s own work is included to illustrate the value and actuality of second quantization. 

We have carried out tins discussion in occupation number representation or coordinate representation each with a definite number N of particles. Similar results follow for the Fock space representation and the properties of grand ensembles. Averages over grand ensembles are also independent of time when the probabilities > are independent of time, whether the observable commutes with H or not. 

Number density operator, total, 452 Number density of particles, 3 Numbers, representation in digital computation, 50 Numerical analysis, 50 field of, 50... 

The operators W, A, occurring above, should be taken in the second-quantization form, free of explicit dependence on particle number, and Tr means the trace in Fock space (see e.g. [10] for details). Problems of existence and functional differentiability of generalized functionals F [n] and r [n] are discussed in [28] the functional F [n] is denoted there as Fi,[n] or Ffrac[n] or FfraoM (depending on the scope of 3), similarly for F [n]. Note that DMs can be viewed as the coordinate representation of the density operators. 

The importance of orbital dependent functionals for a correct representation of the atomic shell structure, the correct properties of v for heteronuclear molecules, and the related particle number dependent properties will be discussed in Sect. 5.5. 

This suggests that in the particle-hole representation each occupied one-particle state in the lN configuration can be assigned a value of the z-projection of the quasispin angular momentum 1/4 and each unoccupied (hole) state —1/4. When acting on an AT-electron wave function the operator a s) produces an electron and, simultaneously, annihilates a hole. Therefore, the projection of the quasispin angular momentum of the wave function on the z-axis increases by 1/2 when the number of electrons increases by unity. Likewise, the annihilation operator reduces this projection by 1/2. Accordingly, the electron creation and annihilation operators must possess some tensorial properties in quasispin space. Examination of the commutation relations between quasispin operators, and creation and annihilation operators... 

A unique feature of the occupation number representation is that the number of electrons does not appear in the definition of the Hamiltonian operator in this form as it does in the wavefunction form. This is because all of the occupation information resides in the bras and kets. This is true for any operator in second quantized form. This feature is used to advantage in theories that allow the number of particles to change, and to a more limited extent in the calculation of electron affinities and ionization potentials. It is less important to the MCSCF method but it is useful to remember that the bras and kets contain all of the occupation information. Other details of the wavefunction, such as the AO and MO basis set information, are included in the integrals that are used as expansion coefficients in the second quantized representation of the operator. 

On the other hand, for systems with low particle-number density and low collision frequencies the estimator h will yield a poor representation of the NDE. Nonetheless, this does not imply that the NDE cannot be defined for such systems. Indeed, it is still precisely defined by Eq. (4.9). Instead it simply states that it will be extremely difficult to estimate the NDF using a single realization of the granular flow. The practical consequence of this statement is that it will be difficult to validate closure models for the terms in the GPBE (using either DNS of the microscale system or experimental measurements) for systems for which the standard deviation of the estimator is large. 

It is interesting to draw a distinction between the two aspects of correlation which we have considered so far in terms of the second-quantization method for systems of N identical Fermi particles. Those methods (which are but a more effective and general way of formulating Cl) rest upon the occupation-number representation given the set of all possible single-particle states (spinorbitals), one builds a complete set of N-particle states. .. > by constructing Slater determinants or... 

It is obvious that 0o cannot serve as a vacuum in the strict sense of the traditional hole-particle formalism, since the valence orbitals in are partially occupied. A straightforward cluster expansion in the occupation number representation from tpo would thus entail two problems (a) there is no natural choice of vacuum to effect a cluster expansion, and (b) the occupation number representation of cluster operators would refer to orbital excitations with respect to the entire oi thus necessitating the considerations of virtual functions which are by themselves combination of functions. If we want to formulate a many-body theory using if>o as the reference function, we need constructs where these cause no problems. 

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