-
7
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-
84951357027
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-
It is here assumed that the approximate wave functions [formula omitted] are normalized to unity. If the bounds of this paper are applied to unnormalized [formula omitted] like the Fourier expansions, we must consider [formula omitted] instead of [formula omitted]
-
-
-
-
10
-
-
0000965806
-
-
especially pp. 266-269. The derivation as given there is restricted to linear variational approximations [formula omitted]
-
(1959)
Adv. Chem. Phys
, vol.2
, pp. 207
-
-
Löwdin, P.O.1
-
12
-
-
0001606288
-
-
For the relation between length and velocity formula in case of “nonlocal potentials,” e.g., the Hartree-Fock potential, see the discussion of
-
(1971)
Phys. Rev. A
, vol.3
, pp. 1242
-
-
Starace, A.F.1
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13
-
-
85034723478
-
-
What we can actually show is the existence of positive numbers [formula omitted] such that [formula omitted] where n and p are state numbers. If this inequality were saturated, the speed for the nth state would be at most as fast as the slowest speed of the lower lying states. A test calculation done for the hydrogen atom using the basis (25) showed, however, that the speed of the first excited state corresponds to expression (32) and thus to Eq. (22), even if the ground state energy converges more slowly.
-
-
-
-
18
-
-
0038227362
-
-
A missing factor of ζ should be inserted on the right-hand sides of Eqs. (33) and (84)
-
(1981)
Adv. Quant. Chem
, vol.13
, pp. 155
-
-
Klahn, B.1
-
20
-
-
85034730695
-
-
Note that Sobolev norms obtained from different values c (e.g., those of Eqs). (22) and (32) are equivalent and induce the same Sobolev space.
-
-
-
-
21
-
-
85034725900
-
-
Note that [formula omitted] and [formula omitted] as defined by Eq. (21) are different functions. Both are best approximations of [formula omitted] however, [formula omitted] in the space [formula omitted] and [formula omitted] in the Sobolev space [formula omitted]
-
-
-
-
25
-
-
84951357032
-
-
Criterion (22), which yields the exact speed, is useless in this case since it requires knowledge of [formula omitted]
-
-
-
-
26
-
-
84951357033
-
-
Hölder’s inequality [formula omitted] [cf. M. Reed and B. Simon, Methods of Modern Mathematical Physics (Academic, New York, 1972), Vol. I, p. 68] gives Eq. (54b), if we choose [formula omitted] and [formula omitted] specifying k to be 3, this equation coincides with Eq. (54b).
-
-
-
-
27
-
-
85034730816
-
-
Whereas the asymptotic factors [formula omitted] and [formula omitted] can be determined exactly if k is even [cf. Eqs. (42), (43), (45), and (53)], in the case of odd k we obtained only lower and upper bounds by means of the interpolation technique: [formula omitted] where the right-hand side holds for [formula omitted] and the left-hand side for [formula omitted]
-
-
-
-
28
-
-
84951357031
-
-
The analogous calculation for the hydrogen ground state suggests also the asymptotic speed exponents of Eq. (60).
-
-
-
-
44
-
-
84951357016
-
-
The expectation values of Tables 2-8 and 9-23 are obtained from normalized wave functions [formula omitted] The exponent [formula omitted] of Table 6 is an empirical value.
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-
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