Primitive approximation

This qualitative discussion cannot be reflected by Eq. (10.47) which is a very primitive approximation especially with respect to intermediate Reynolds number 1 < Re < 100. 

Figure 4.20a Application of the Jacobi transformation and the canonical orthonormalization procedure to the calculation of the Is orbital energy in hydrogen using the Dunning (4s 2s) Gaussian basis (46). Note that the calculation returns a second set of coefficients in cells H 22 and H 23, which are simply a product of the calculation procedure and give rise to a primitive approximation to the 2s orbital (22).

Figure 4.20b Comparisons of the exact radial projections of the hydrogen Is and 2s radial functions with those of the two linear combinations resulting from the calculations in fig4-20ab.xls. The second linear combination is a primitive approximation to the hydrogen 2s radial function to the extent that it is orthonormal to the Is function [cells canonical K 24 to SM 24].

This is called the primitive approximation [1]. More accurate approximation can also be utilized [18]... 

It is worth noting that within the PI implementation, we are mainly interested in evaluating the trace of the density matrix, as it is directly related to the partition function. Also when using the primitive approximation, we neglect terms that are of the order x. To improve the precision of results in MC simulations and to achieve faster convergence as P increases, higher order corrections (or propagators of the density matrix) have been developed. 

In this section, we will show how the thermal density matrix is used in PIMC to compute quantum viiial coefficients. Consider the Hamiltonian of a monatomic molecule like helium with mass m (Eq. 7). Using the primitive approximation (Eq. 4), Trotter formula (Eq. 5), and following the procedure outlined in Ref. [9], we can obtain the kinetic-energy operator matrix elements as ... 

Also, and for its intrinsic interest in further developments, it is convenient to give the standard definition of the so-called centroid (i.e., the CM) of the P necklace describing particle j in the primitive approximation... 

The primitive approximation contains all the physics involved in the treatment of quantum many-body systems at nonzero temperature. It is simple, intuitive, and highly flexible. Moreover, it reveals clearly the so-called classical isomorphism, Eqs. (25)-(27), a correspondence that has important consequences on a range of different issues (e.g., formal study of structures and computational techniques). Nevertheless, the computational efficiency of the primitive scheme is generally poor as pointed out in earlier applications. A couple of examples will help to understand these drawbacks. First, the kinetic energy, given by the first two terms on the right-hand side of Eq. (31), which shows increasing variances with P, a fact associated with the stiffness of the harmonic links in [63]. Second, the P con-... 

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