2 single harmonic oscillator (everything generalizes imme- diately to multiple oscillators at different lattice sites) the Fock states with fixed boson number are the states | n 〉 = ( b $^{†}$) n $_{/}$√$_{n}$$_{!}$ | vac 〉 . If a force, say − √ ϵ$_{p}$ω$_{0}$ ( b † + b ), is applied the average number of bosons in the groundstate is given by g 2 = ϵ$_{p}$/ω$_{0}$ . The larger ϵ$_{p}$ or the smaller ω$_{0}$ , the more bosons are needed to account for the os- cillator elongation 〈 b † + b 〉 = 2 g . On the other hand, we know that the groundstate is simply a coherent state | g 〉 c = ˆ S ( g ) | vac 〉 , obtained from the vacuum with the shift operator ˆ S ( α ) = exp( αb † − α $^{∗}$b ). For the polaron problem, the gain in efficiency ob- tained using SOS is demonstrated in Tab. I. Only this efficiency gain allows us to perform calculations for fre- quencies ω$_{0}$ ≪ 0 . 1 at λ = 1, extending the accessi- ble parameter range by more than one order of magni- tude into the adiabatic regime. The present calculations, down to ω$_{0}$ = 0 . 01, require N$_{gen}$ ≤ 23, and could all be performed on small desktop computers. The valid- ity of our implementation was checked against calcula- tions from Diagrammatic Quantum Monte Carlo (data for the 3D polaron mass at ω$_{0}$ = 0 . 5 were provided by A. S. Mishchenko). The main restriction of the VED construction, com- mon to most ED studies, is the large number of phonons at large couplings or small phonon frequencies. To overcome this restriction we will replace the standard Fock basis with a more general construction. For a VED is based on an increasing sequence of subspaces of the complete Hilbert space. A certain subspace con- tains all basis states that are produced through a given number N$_{gen}$ of applications of the Hamiltonian to the phonon vacuum, that is either by excitation of a phonon or an electron move. For example, it contains the state with N$_{gen}$ phonons at the electron position, and the one with a single phonon N$_{gen}$ − 1 lattice sites away from the electron. The number of elements in the subspaces grow only by a factor close to D +1 as N$_{gen}$ increases by 1, and already small subspaces provide a good approximation of the true groundstate. All quantities, e.g. variational es- timates of the groundstate energy, can be improved by increasing N$_{gen}$ until convergence. The average elongation and energy of the SOS | n 〉 σ grows with n , so that they can replace the Fock states in the VED. The optimal parameter σ can be determined variationally through minimization of the groundstate energy. For our purposes, the choice σ = g/N$_{gen}$ is close to optimal, and was used in all calculations. For larger coupling g , hence larger optimal σ , the SOS account for larger oscillator elongations. Conversely, if the maximal number of states N$_{gen}$ per oscillator is increased and σ de- creases, the SOS sample ever finer details of the oscillator wavefunctions. In fact, since in our calculations always σ ≲ 1, the states | nσ 〉 c overlap in position space, and the SOS do not resemble coherent states after orthogonaliza- tion. Instead they interpolate between ‘quantum’ Fock states generated by b † and ‘classical’ coherent states gen- erated by ˆ S ( α ). Note that a construction with a fixed small number of coherent states would have to fail: For small ω$_{0}$ it is not possible to represent all oscillator wave- functions – with small width and different positions – if only few states are available. For the SOS the necessary increase in accuracy is accomplished by construction if N$_{gen}$ grows. Our calculations for the Holstein model are performed with standard ED methods, such as the Lanczos algo- rithm and the Kernel Polynomial Method $^{40}$. The fun- damental task is the construction of an appropriate sub- space of the complete infinite Hilbert space of the single fermionic and many bosonic degrees of freedom of the Hamiltonian (1). A very efficient construction is provided by VED, which achieves extreme accuracy not only for the polaron groundstate, especially in the regime of in- termediate coupling and phonon frequency $^{19}$, but also for models with different types of fermion-boson coupling $^{41}$. The fermionic operators c † n create the electron, and the bosonic operators b † n create a phonon at site n . The sums run over all lattice sites n , with the vectors e i along the lattice axes. The Holstein model has three param- eters: the electron transfer integral (or hopping matrix element) t , the phonon frequency ω$_{0}$ , and a third pa- rameter ϵ$_{p}$ , λ = ϵ$_{p}$/ 2 Dt , or g 2 = ϵ$_{p}$/ω$_{0}$ specifying the electron-phonon coupling strength. In the following we give all numerical values with the hopping matrix ele- ment as the unit of energy, so that t = 1. All results are for a single electron at zero temperature. The Hamiltonian of the Holstein model on a chain (cu- bic lattice) in dimension D = 1 ( D = 3) is given by the 1D case is continued with the f-sum rule and optical conductivity in Sec. V. We conclude in Sec. VII with a discussion of our results in the broader context of polaron formation under general conditions. II. THE HOLSTEIN MODEL AND THE NUMERICAL METHOD H = − t D ∑ i =1 ∑ n c † n + $_{e$_{i}$}$c$_{n}$ + c † $_{n}$c$_{n}$$_{+}$$_{e}$$_{i}$ − √ ε$_{p}$ ω$_{0}$ ∑ n ( b † n + b n ) c † n c n + ω$_{0}$ ∑ n b † n b n . (1)