Article pubs.acs.org/JPCA
Anharmonic Vibrational Treatment Exclusively in Curvilinear Valence Coordinates: The Case of Formamide Published as part of The Journal of Physical Chemistry A virtual special issue “Spectroscopy and Dynamics of Medium-Sized Molecules and Clusters: Theory, Experiment, and Applications”. F. Richter,*,† F. Thaunay,†,§ D. Lauvergnat,‡ and P. Carbonnière† †
Groupe de Chimie Théorique et Réactivité IPREM-ECP, Université de Pau et des Pays de l’Adour, Pau 64000, France Laboratoire de Chimie Physique (UMR 8000), Université Paris-Sud, Orsay 91405, France
‡
ABSTRACT: A highly correlated approach using curvilinear valence coordinates is applied to calculate the vibrational fundamentals and some combination modes of the formamide molecule with high accuracy. A series of potential energy surfaces (PESs) has been generated by AGAPES, a program for adaptive generation of adiabatic PESs, at various electronic structure qualities until excellent nonaccidental agreement with the experimentally assigned fundamental transitions was reached at the CCSDT(T)-F12a/aug-cc-pVTZ level of theory using the improved relaxation method of the Heidelberg multiconfiguration time-dependent Hartree (MCTDH) package in connection with an exact expression for the kinetic energy in valence coordinates generated by the TANA program. By comparison of the overtone series ν1−3ν1 to experiment, we demonstrate that the known problems concerning the floppy ν1 wagging motion are solved within this approach. The potential energy coupling as well as the vibrational coupling in curvilinear coordinates is discussed together with the efficiency of this approach.
1. INTRODUCTION The computation of accurate gas phase mid-infrared spectra is still a nonroutine task for larger than four atomic molecules in the presence of strong anharmonic couplings or large amplitude motions. Both normal1−4 and curvilinear coordinate strategies5−10 are being followed by modern computational methods (see ref 11 for an overview). Although it is commonly agreed that curvilinear coordinates may lead to significantly improved separation, of both the potential energy surface (PES) and the vibrational motion, their application was hindered by the lack of representations of the kinetic energy operators (KEOs) suited for efficient numerical application. This obstacle has been overcome after more than a decade’s research. Today, the KEO can be exactly represented either as a numeral on each given grid point12−18 for any kind of curvilinear coordinates or in the form of a compact analytic expression for a particular group of curvilinear coordinates.9,19−24 Notably, upon the very recent introduction of subsystems to the polyspherical approach of TANA,25,26 that group includes now all practical types of valence coordinates. The automatically generated KEO’s of TANA have a sum of product (SOP) form, a mandatory feature for many efficient integration schemes within the vibrational calculation step. In particular, TANA generates automatically the KEO in a valid multiconfiguration time-dependent Hartree (MCTDH)27−33 operator format. Hence, today one is in the position to establish a program suite for infrared spectra calculations that fully © XXXX American Chemical Society
exploits the advantages of valence coordinates without concessions to efficiency or accuracy due to an unsuited form or due to an approximate character of the KEO expression, respectively, given the fact that efficient PES generation as well as efficient vibrational integration routines are readily available for valence coordinates. In this work we will combine AGAPES,34,35 a program for adaptive generation of adiabatic PESs, and improved relaxation11 of MCTDH30,32 for these tasks. AGAPES is specifically designed for automatic generation of adiabatic valence coordinate PESs in SOP form for large molecules showing strong anharmonic couplings. It explicitly exploits the locality and the excellent separation of the potential energy in valence coordinates to achieve favorable scaling.35 Hence, in combination with TANA, compact SOP Hamiltonians in valence coordinates can be generated automatically and for any type of molecule. Improved relaxation of MCTDH, essentially a multiconfiguration self-consistent field (MCSCF) approach, is reliable and efficient for medium-size systems, and calculations on up to 21-dimensional (21D) systems36−39 have been reported. Systematic errors are much less likely than one might expect for VSCF schemes including perturbative treatment of correlation. Very recently, a slightly modified version improved relaxation in connection with a complex Received: September 1, 2015 Revised: November 4, 2015
A
DOI: 10.1021/acs.jpca.5b08482 J. Phys. Chem. A XXXX, XXX, XXX−XXX
Article
The Journal of Physical Chemistry A Table 1. Symbolic Z-Matrix with Equilibrium Data from CCSD(T)-F12/avtz Calculationsa atom 1
atom 2
stretch
atom 3
angle
atom 4
dihedral
C1 N2 O3 H4 H5 H6
C1 C1 C1 N2 N2
R1(1.359) R2(1.214) R3(1.104) R4(1.004) R5(1.006)
N2 N2 C1 C1
A1(124.48) A2(112.64) A3(121.07) A4(119.21)
O3 O2 O1
D1(180) D2(180) D3(0)
The first two, three, and four atoms in each row form a stretch, an angle, and a dihedral angle, respectively. The equilibrium values at the CCSD(T)-F12/avtz level are given in brackets in (Å) and degree.
a
N
absorbing potential has been successfully applied to resonance states of the formyl radical.40 Calculations on the dynamics of larger systems are feasible41 with the multilayer multiconfiguration time-dependent Hartree (ML-MCTDH).42,43 It has been shown recently44 that ML-MCTDH starts to be more efficient than the single-layer version of MCTDH already at 9D. Although ML-MCTDH has not yet been used in connection with improved relaxation, and we will use single layer MCTDH with mode combinations throughout in this work, it is a viable option for our future applications. Another important feature of MCTDH is that it is ready to work with any SOP operator whereas many other programs are confined to just one type of coordinates. Formamide is an interesting prototype molecule. The most recent experimental mid-infrared assignment has been done by McNaughton et al.45 Structure and dynamics of large proteins are monitored using the amide vibrational modes as fingerprint of that conformation, and formamide is the simplest molecule that combines the characteristic HNCO (peptide) and the amino group, albeit, being planar, which is atypical46 for larger proteins. At least the amide-I mode is found to be very sensitive to underlying secondary structures.47,48 What makes formamide interesting for a fully curvilinear approach test molecule is its shallow out-of-plane PES for the wagging mode together with the fact that the NH stretches of formamide are heavily coupled to this one and other low frequency motions in the normal mode picture. Valence coordinates largely reduce the potential energy coupling as well as the vibrational motion correlation and, hence, allow for accurate results at low computational costs. For instance, the wagging motion may be well represented by one single symmetrized torsional coordinate. The impact to accuracy of using normal coordinates vs valence coordinates for vibrational integration schemes at various correlation levels for formamide has been addressed by Bounouar et al.49 using approximate expressions of KEOs in the curvilinear case. Therefore, we will also discuss our results obtained by using an exact KEO expression in view of their findings.
V (q1 ,..,qN ) =
N
N
∑ Vi(1) + ∑ Vij(2) + ∑ i=1
... +
i
