J . Phys. Chem. 1990, 94, 3944-3951
3944
vibration are unaltered. We would like to have the Cartesian components of fit) and fiB)and of Sfic) and SfiB) in the space-fixed coordinate system X,Y,Z. The total force on BC, the force along the BC bond, and the torque are most easily derived in the molecule-fixed coordinate system, x,y,z. Since the total force on BC is to be unchanged, ticc)+ 6cB) = 0. We then simplify notation and write 6fiC)
=
f,
8f(B) = -f
(AI)
If C is a unit vector from C to B such that C = rBc/rBc,then the requirement that the force along BC is unchanged means that (fit) + &fiC)).C - (fiB) + af(B)).C = f(c).$- f(B).C, or
f-C = 0
which takes any vector q in the molecular system to the vector Q in the space-fixed coordinate system, Q = nq. ll can be represented by using the Euler angles 0 (the angle between 6 and the Z axis), 4 (the angle between the projection C on the XY plane and the negative Y axis), and # (the angle between the x axis and the normal to the 2 - z plane). cos $ cos 4 - cos 0 sin $ sin II = cos $ sin $I cos R cos 4 sin $ sin 0 sin 6
+
[
+
This result is
[%]
(A21
Finally, the total torque on BC due to the sum of the original and additional forces is set to zero
+ arcB)) - pBTBC x (fit) + 6c") = 0 (A3) where pB = m e / ( m B+ mc) and p C / ( m B+ mc). Dividing eq A3 pCrBC
-sin $ cos 4 - cos 0 sin @ cos $ -sin $ sin $I + cos R cos 4 cos $ sin 0 cos $
= pcrI[
- pBII[
sin 0 sin
@
I
-sin R cos 4 (A6)
cos R
F]
(A7)
x (fin
by rBc and using eq A I , we get the following implicit equation for f $
x (pcfiB'- pBfic') = 5 x f
(A41
which can be solved in the coordinate system (x,y,z) where z is in the direction C and x and y are perpendicular to each other and to z. From eq A2, f, = 0 andf, andf, can be obtained from eq A4 by taking the dot product with unit vectors in the direction of y and x, respectively, which givesf, = P(C~,(~)- ~,af,(~)andf, = p(Cfy(B) - pafy(c). The solution for f can be written in matrix form:
This is the solution to the problem for the additional forces in the molecular frame, when the original forces are also given in the molecular frame. For simulation in a space-fixed coordinate system, we need to transform all forces to the space-fixed frame in which the components of the vector C are (Zx,ty,t,).This can be done by multiplying eq A5 by the transformation matrix ll
We now need to express the right-hand side of eq A6 in terms of the input forces F(c) and F(B)in the space-fixed frame, using f,(') = [n'F(B)]x andf,(B) = and similarly for FCC),llt being the transpose of n. The final result after performing the matrix multiplications is
F = I'(pCF(B)- pBF(C))
(A81
in which r is given below. This force F should be added to the force on atom C and subtracted from the one on B at every time step. The matrix elements of I' are bilinear combinations of the elements of the matrix ll. When these elements are expressed in terms of the Euler angles, the dependence of the elements on the angle drops out and they can be expressed in terms of the components of 0 in the space-fixed frame, (Zx,ty,Pz) = (sin 0 sin 4, -sin 6' cos 4, cos 6')
+
[
1-
cx2
r = -2dY -a$,
-2dy
-2&
1-2:
-2gZ
-2gz
1
- ZZ2
]
('49)
Note that the vector C is calculated in the normal integration algorithm, so that very little extra computation is required to implement the above method.
Theoretical Study of the Excitation Energies of Methaniminium Cation, Propeniminium Cation, and Propenimine Ping Du, Stephen C. Racine, and Ernest R. Davidson* Department of Chemistry, Indiana University, Bloomington, Indiana 47405 (Received: August 22, 1989)
Ab initio calculations on the ground states and excited states of methaniminium cation, propeniminiumcation, and propenimine were performed. Ground-state population analyses show that the positive charge for the cations is largely located on the carbon atom next to the nitrogen. No low-lying 2'A,-like state was found for methaniminium, whose first singlet excited state is a u K* excitation. Although the ZIA,-like state of propenimine was calculated to be only slightly higher than the 'B,-like state, protonation of this molecule strongly stabilizes the ionic 'B,-like state. To test the reliability of these results, calculations on the excitation energies of butadiene and acrolein were also performed with the same basis sets and method. The resulting excitation energies for the singlet states are compared with the available experimental values.
-
The electronic properties of the lowest lying excited state of the protonated Schiff base of retinal (PSBR), the chromophore of the visual pigment Rhodopsin, provide important information toward understanding the primary event of visi0n.I Because of the similarity between the molecular structures of PSBR and ( 1 ) Sandorfy, C . ; Vocelle, D. Can. J . Chem. 1986, 64, 2251.
0022-3654/90/2094-3944$02.50/0
polyenes, this question has historically been related to the doubly excited 2'A, state, which was found to be the first excited state of long-chain polyenes in condensed phases.2 Although it has been shown that the unprotonated Schiff base of retinal (SBR) (2) Hudson, B. S.; Kohler, B. E.; Schulten, K. Excited States; Lim, E. C., Ed.: Academic Press, New York, 1982; Vol. V I , p I .
0 1990 American Chemical Society
The Journal of Physical Chemistry, Vol. 94, No. 10, 1990 3945
Excitation Energies of Imine Compounds
TABLE I: RHF/DZP Optimized Geometrical Parameters of Methaniminium, Propenimine, and Propeniminium geom param'
R2 1 R32 R43 RS I R6 1 R72 R83 R94 R104 A321 A432 A512 A612 A723 A832 A943 A I043
trans-
CHZNH2' 1.298 1.073 1.007 1.007 1.073
119.4 121.3 121.3 121.1
CHZCHCHCHP 1.343 1.467 1.343 1.094 1.094 1.094 1.094 1.094 1.094 122.8 122.8 119.5 119.5 119.5 119.5 119.5 119.5
CH2CHCHNH2+ CH,CHCHNH Bond Length, A 1.288 1.260 1.438 1.476 1.328 1.339 1.001 1.005 1.001 1.086 1.077 1.075 1.076 1.076 1.078 1.075 1.075 Bond Angle, deg 123.766 119.602 121.363 121.494 119.578 118.777 122.378 120.652
121.414 122.061 111.517 115.338 116.182 121.415 121.549
cisCHzCHCHNH
CHZCHCHO'
1.259 1.48 1 1.328
1.219 1.470 1.345
1.007 1.082 1.078 1.077 1.075
1.084 1.086 1.086
126.712 122.379
123.267 119.833
1.108
1 1 1.844 115.746 117.058 121.448 121.693
115.100
117.500 119.766 121.450
'The definitions of the geometry parameters are given in Figure 1. bExperimental values taken from ref 15. 'Experimental values taken from ref 16
also has the 2IA,-like state as the lowest excited state,) this energy order seems not to apply to PSBRS3" Because the Schiff base of retinal is p r o t ~ n a t e din ~ ,the ~ biological environment, the singly excited IB,-like state should be the state formed immediately after the absorption of a photon. As the first part of the study using ab initio theory, we have carried out calculations on the excitation energies of methaniminium (CH2=NH2+), propeniminium (CH2=CH-CH=NH2+) cations, and propenimine (CH2=CH-CH=NH), in order to test the basis sets and methods to be used in the calculation of PSBR. Protonation of PSBR not only plays a critical role in determining the lowest lying excited state, it also creates charge centers along the conjugated chain next to which counterions from the charge centers of the protein are possibly l o ~ a t e d . ~ .These ~ counterions regulate the absorption wavelength of rhodopsin.lOJ1 Charge distributions were obtained by performing a population analysis of the ground states of methaniminium, propeniminium, and propenimine at the RHFSCF level. Since there is only one double bond in methaniminium, there exists no low-lying doubly excited state for this molecule. Although neither propeniminium nor propenimine possesses C2h symmetry, conventionally the symmetry labels of butadiene are used to describe the corresponding singly and doubly excited states of propeniminium and propenimine. In this paper, a different kind of notation is adopted. The 'B,-like state of propeniminium, for example, is labeled as ' ( ~ 2 , ~ 3 *indicating ), a singlet state dominated by exciting a single electron from the r2orbital to the T ~ * orbital. Similarly, the 2IA,-like state is nominally labeled as ' ( ~ 1 , ~ 3 *although ), the configuration shown is only one of several important configurations in the true wave function of this state. Additionally, symmetry labels are also used for these states, such as 2IA' for the IB,-like state of propeniminium.
Geometries and Basis Sets Geometry optimizations of methaniminium cation, propeniminium cation, and propenimine were carried out at the R H F level, ( 3 ) Murry, L. P.; Birge, R. R. Can. J . Chem. 1985, 63, 1967. (4) Birge, R. R.; Murry, L. P.; Pierce, B. M.; Akita, H.; Balogh-Nair, V.; Findsen, L. A.; Nakanishi, K. Proc. Natl. Acad. Sci., U.S.A. 1985, 82, 41 17. (5) Becker, R. S.; Freedman, K. J . Am. Chem. SOC.1985, 107, 1477. (6) Freedman, K. A.; Becker, R. S. J . A m . Chem. SOC.1986, 108, 1245. (7) Honig, B.; Ebrey, T.; Callender, R. H.; Dinur, V.; Ottolenghi, M. Proc. Natl. Acad. Sci. U . S . A . 1979, 76, 2503. (8) Aton, B.; Doukas, D.; Narva, D.; Callender, R. H.; Dinur, U.; Honig, B. Biophys. J . 1980, 29, 79. (9) Blatz, P. E.; Mohler, J. H. Biochemistry 1972, 11, 3240. ( I O ) Blatz, P. E.; Mohler, J. H. Biochemistry 1975, 14, 2304. (11) Kakitani, H.; Kakitani, T.; Rodman, H.; Honig, B. Photochem. Photobiol. 1985, 4 1 , 471.
H6
H8
I
H7
H6
I
H9
I
H7
Figure 1. Atom numbering for geometry definitions. X is C for butadiene, N for propenimine and propeniminium, and 0 for acrolein. The existence of H5 or H6 depends on the molecule.
with the standard Dunning double-{12 basis set plus polarization functions (DZP) that are provided in the HONDO package of programs.13 C2, symmetry was constrained for methaniminium. Propeniminium and propenimine were constrained to be planar. Both isomers of propenimine, cis and trans, were optimized. The optimized geometry parameters are listed in Table I with definitions shown in Figure 1. The geometry found for methaniminium is very similar to a theoretical geometry reported previo~sly.'~For comparison, the experimental geometry parameters of butadiene15 and acrolein16 are also included in Table I. The optimized geometries of cis and trans-propenimine are very similar except for a few parameters. The C2-C3 bond length (1.476 A) of trans-propenimine is 0.005 A shorter and the N1C2
