# Variational Monte Carlo Calculations of {sup 3}He and {sup 4}He with a Relativistic Hamiltonian - II Page: 4 of 7

where

8rnp(Pij) = I2 (7a - (7) x ik (V i6,i2) , (5)
and 6vQM(Pik) contains terms that come from the commutator of (a, - nj) with the spin
operators in ieij. The 6VTP(Pij) originates from the classical Thomas precession [6,7]. The
precession of the spin si in the frame moving with velocity Pi/2m is given by -Vi3ii x
Pi /4m2 up to order 1/m2. Thus the Thomas precession potential for particle i is:
1 VigJ x Pik 1
- - - 4 Pn2i T i(vi,) . (6)
Both particles have same velocity due to their center of mass motion, but their accelerations
due to ig are equal and opposite. Therefore the Thomas precession potential for the particle
j is -a x Pik - (Vigji ) /8m2, and together with (6) it makes up the 6VTP(Pij). After some
algebra we obtain:
5VTP(Pig) 2 [(v' - v' + v' + 3) P r x (ni - nj)
-i 2v' + v' + 3- (P - o r -o2 - P - on r - an) + ri -r term, (7)
where v' denotes wOv/Or, the ij subscripts of r, P and v, are omitted for brevity, and the
Ti -* term has vT, vUT and vtT in place of v,, vQ and vt.
The 6vQM(Pik) does not have a classical analogue; it is found to be:
65VQM(Pj) 22 (Vt - v) (P " O, 0' " V - P " 0' O, V)
3i Vt
- 4r2 - P r (n,.r n.V- (7jr (7iV)
3i Vt
4m2 2 (P.nr.0 -P cr
+Ti - Tj terms (8)
from eq. (4).
It is convenient  to express 6v(Pi) given by eq. (2) as:
by(Pig) = 6VRE(Pij) - 6VLC(Pij) - 6VTP(Pij) + 6VQM(Pij)- (9)
4

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