Efficient “Middle” Thermostat Scheme …
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3.4 Multi-electronic-state PIMD
Consider a Hamiltonian with N s electronic states of the form ˆ
H = ˆ
T + ˆ
V, where
ˆ
V = V( ˆ
R) is a N s × N s symmetric diabatic potential energy matrix and ˆ
T is the
kinetic energy matrix. The canonical partition function is defined as
Z = Tr ne [e
−β ˆ
H
]
(51)
and a specific physical property of interest is evaluated by
ˆ
B
=
1
Z
Tr ne [e
−β ˆ
H ˆ
B]
(52)
In Eqs. (51) and (52) the trace is integrated over both the nuclear and electronic
degrees of freedom, i.e.
Tr e [ ˆ
O] =
N s
n=1
n| ˆ
O|n
(53)
and
Tr n [ ˆ
O] =
dRR| ˆ
O|R.
(54)
Substituting the resolution of the identity into Eq. (51) yields
Z = lim
P→∞
PM
2πβ 2
P/2
dR 1 . . . dR P exp
−
β
2
ω
2
P
P
i=1
(R i − R i+1 )
T M(R i − R i+1 )
× Tr e
P
i=1
O
T
(R i )O(R i )
(55)
where O(R i ) is related to the splitting scheme. We have studied three typical
decomposition schemes, namely, the “diagonalization”, “first-order expansion”, and
“hyperbolic function” methods in Ref. [40].
Because Tr e
P
i=1 O
T
(R i )O(R i )
is not always positive-definite for general
multi-electronic-state (MES) systems, regardless of which decomposition scheme is
employed, it is not recommended to use either Tr e
P
i=1 O
T
(R i )O(R i )
or its absolute value to define an effective potential function φ(R 1 , . . . , R P ) for performing
PIMD. A reasonable effective potential is defined by