= Solution
Model "lme2" replaces $b_j$ by a random intercept and random PTHK slope:
$$
\operatorname{THK}_{ij}
=x_{ij}^T\beta+b_{0j}+b_{1j}\operatorname{PTHK}_{ij}
+\varepsilon_{ij},
$$
with a fitted bivariate normal covariance matrix for $(b_{0j},b_{1j})$. This adds a random-slope variance and an intercept-slope covariance.
The likelihood-ratio statistic is
$$
2\{-2680.4-(-2684.6)\}=8.4.
$$
An ordinary chi-squared reference is unreliable because the null random-slope variance is on the boundary and its correlation is unidentified there; the reported correlation of one also signals a nearly singular fit. A valid practical test is a <parametric bootstrap>: simulate many datasets from fitted "lme1", refit both models by maximum likelihood to each, recompute the likelihood-ratio statistic, and estimate the p-value by the fraction at least $8.4$. The AIC favors "lme2" slightly, whereas its BIC is larger, so the descriptive criteria do not agree.
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