14 Simulating Quantum Dynamics in Classical Nanoscale Environments
531
E 1
E 0
Q m
q m
J
J
J
J
J
Fig. 14.5 One-dimensional chain model of H-bonded peptide groups (blue spheres) in an alphahelical polypeptide. The position of the mth amide I mode (green rectangles) is denoted by q m ,
while the longitudinal displacement of the mth unit (blue double-sided arrows) is denoted by Q m .
Initially, the first amide I mode in the chain is vibrationally excited (as indicated by the red arrow)
with energy E 1 , and the remainder of the amide I modes are in their ground states with energies
E 0
where |φ m = |χ m1
n =m |χ n0 is the mth singly excited state and |0 =
m |χ m0
is the ground state of a chain of amide I modes. (N.B.: |χ m0 and |χ m1 denote
the ground and first excited states, respectively, of the mth amide I mode). The
ground-state matrix element is given by H 0 = T + E 0 (Q), where T = i
P 2
i
2M is the
kinetic energy associated with the motion of the H-bond lattice displacements and
the potential energy has the following form:
E 0 (Q) =
m
E m0 + (1 − δ m,1 )
W
2
(Q m − Q m−1 )
2
+ w m0 (Q)
.
(14.27)
The excited state matrix elements are given by
H mn = δ m,n [H 0 + E m1 − E m0 + w m1 (Q) − w m0 (Q)]
−J
δ m−1,n m |q m q m−1 |φ m−1
+ δ m,n−1 n |q n q n−1 |φ n−1
,
(14.28)
where δ i,j is the Kronecker delta function, J is the parameter governing the strength
of the transition dipole moment coupling between neighbouring amide I modes,
W is the force constant governing the motion of the H-bond lattice displacements,
E m1 − E m0 is the energy gap between the ground and first excited states of the
mth amide I mode, q m is the position of the mth amide I mode, and Q = {Q m }
corresponds to the set of H-bond lengths. The term w mν (Q), which describes the
coupling between the H-bond displacements and the amide I modes, is given by
w mν (Q) = (1 − δ m,1 )
χ
2
(Q m − Q m−1 ) mν |q
2
m |χ mν
(14.29)
where χ is the coupling strength. The values of the various matrix elements of q m
and q 2
m are taken from Table 2 in Ref. [73], while the values of the parameters
531
E 1
E 0
Q m
q m
J
J
J
J
J
Fig. 14.5 One-dimensional chain model of H-bonded peptide groups (blue spheres) in an alphahelical polypeptide. The position of the mth amide I mode (green rectangles) is denoted by q m ,
while the longitudinal displacement of the mth unit (blue double-sided arrows) is denoted by Q m .
Initially, the first amide I mode in the chain is vibrationally excited (as indicated by the red arrow)
with energy E 1 , and the remainder of the amide I modes are in their ground states with energies
E 0
where |φ m = |χ m1
n =m |χ n0 is the mth singly excited state and |0 =
m |χ m0
is the ground state of a chain of amide I modes. (N.B.: |χ m0 and |χ m1 denote
the ground and first excited states, respectively, of the mth amide I mode). The
ground-state matrix element is given by H 0 = T + E 0 (Q), where T = i
P 2
i
2M is the
kinetic energy associated with the motion of the H-bond lattice displacements and
the potential energy has the following form:
E 0 (Q) =
m
E m0 + (1 − δ m,1 )
W
2
(Q m − Q m−1 )
2
+ w m0 (Q)
.
(14.27)
The excited state matrix elements are given by
H mn = δ m,n [H 0 + E m1 − E m0 + w m1 (Q) − w m0 (Q)]
−J
δ m−1,n m |q m q m−1 |φ m−1
+ δ m,n−1 n |q n q n−1 |φ n−1
,
(14.28)
where δ i,j is the Kronecker delta function, J is the parameter governing the strength
of the transition dipole moment coupling between neighbouring amide I modes,
W is the force constant governing the motion of the H-bond lattice displacements,
E m1 − E m0 is the energy gap between the ground and first excited states of the
mth amide I mode, q m is the position of the mth amide I mode, and Q = {Q m }
corresponds to the set of H-bond lengths. The term w mν (Q), which describes the
coupling between the H-bond displacements and the amide I modes, is given by
w mν (Q) = (1 − δ m,1 )
χ
2
(Q m − Q m−1 ) mν |q
2
m |χ mν
(14.29)
where χ is the coupling strength. The values of the various matrix elements of q m
and q 2
m are taken from Table 2 in Ref. [73], while the values of the parameters