  The spin-Peierls transition at $T_{SP}$ of spin-$1/2$ chains with isotropic
exchange interactions has previously been modeled as correlated for $T >
T_{SP}$ and mean field for $T < T_{SP}$. We use correlated states throughout in
the $J_1-J_2$ model with antiferromagnetic exchange $J_1$ and $J_2 = \alpha
J_1$ between first and second neighbors, respectively, and variable frustration
$0 \leq \alpha \leq 0.50$. The thermodynamic limit is reached at high $T$ by
exact diagonalization of short chains and at low $T$ by density matrix
renormalization group calculations of progressively longer chains. In contrast
to mean field results, correlated states of 1D models with linear spin-phonon
coupling and a harmonic adiabatic lattice provide an internally consistent
description in which the parameter $T_{SP}$ yields both the stiffness and the
lattice dimerization $\delta(T)$. The relation between $T_{SP}$ and
$\Delta(\delta,\alpha)$, the $T = 0$ gap induced by dimerization, depends
strongly on $\alpha$ and deviates from the BCS gap relation that holds in
uncorrelated spin chains. Correlated states account quantitatively for the
magnetic susceptibility of TTF-CuS$_4$C$_4$(CF$_3$)$_4$ crystals ($J_1 = 79$ K,
$\alpha = 0$, $T_{SP} = 12$ K) and CuGeO$_3$ crystals ($J_1 = 160$ K, $\alpha =
0.35$,
  $T_{SP} = 14$ K). The same parameters describe the specific heat anomaly of
CuGeO$_3$ and inelastic neutron scattering. Modeling the spin-Peierls
transition with correlated states exploits the fact that $\delta(0)$ limits the
range of spin correlations at $T = 0$ while $T > 0$ limits the range at
$\delta= 0$.
